LaTeX - A Crash Course

Jai Ganesh · 2023-02-04 17:45:36

Square Pyramid

A square pyramid characterized by a square base is a three-dimensional shape having five faces, thus called a pentahedron. The most famous example of such a square pyramid is the Great Pyramid of Giza. A pyramid is a polyhedron that has a base and 3 or greater triangular faces that meet at a point above the base (the apex). Interestingly, pyramids are named after their base, such as

* Rectangular pyramid
* Triangular pyramid
* Square pyramid
* Pentagonal pyramid
* Hexagonal pyramid

Here, we will explore the concept of a square pyramid and its properties. We will discuss different types of square pyramids along with their formula and the net of the square pyramid for better visualization of its figure. We will solve various examples based on the concept for a better understanding.

What is a Square Pyramid?

A square pyramid is a three-dimensional geometric shape that has a square base and four triangular sides that are joined at a vertex. It is a polyhedron (pentahedron) with five faces. A square pyramid consists of a square base and four triangles connected to a vertex. Its base is a square and the side faces are triangles with a common vertex.

A square pyramid has three components.

The top point of the pyramid is called the apex.
The bottom square is called the base.
The triangular sides are called faces.

Properties of a Square Pyramid

Let us list out the properties we have explored in the above image. All these properties are derived from the definition of a pyramid.

* It has 5 faces.
* It has 4 side faces that are triangles.
* It has a square base.
* It has 5 vertices.
* It has 8 edges.

Types of Square Pyramids

We can distinguish the square pyramids on the basis of the lengths of their edges, position of the apex, and so on. Let us discuss the different types of square pyramids.

Right square pyramid

If the apex of the square pyramid is right above the center of the base, it forms a perpendicular with the base. Such a square pyramid is called the right square pyramid.

Oblique square pyramid

If the apex of the square pyramid is not aligned right above the center of the base, the pyramid is called an oblique square pyramid.

Equilateral square pyramid

If all the triangular faces of a square pyramid have equal edges, then the square pyramid is called an equilateral square pyramid.

Square Pyramid Formula

There are formulas for square pyramids for finding the volume, height, base area, and surface area. Here you can see the formulas of the volume, total surface area (TSA), and lateral surface area (LSA) of the square pyramid.

Base Area of a Square Pyramid

Since the square pyramid has a square base, we can calculate its base area using the same formula as the area of square, which is

{side} \times {side}

or

{base \ edge}^2

gives

or .

Volume of a Square Pyramid

The formula to determine the volume of a square pyramid is:

V = \dfrac{1}{3}a^2{h}

gives

.

Here, a is the length of the base and h is the perpendicular height.

Surface Area of a Square Pyramid

There are two types of surface areas, one is TSA (Total Surface Area), and the other is LSA (Lateral Surface Area). When we talk about its surface area, we generally refer to its total surface area (which is the sum of areas of all faces), whereas the lateral surface area is the sum of the areas of the side faces only. Consider a square pyramid of base edge 'a', height 'h', and slant 'l'.

The formula to calculate the surface area of a square pyramid when its height h and base edge a are given:

Curved Surface Area:

2a\sqrt{\left(\dfrac{a^2}{4} + h^2\right)}

gives

or

2al

gives

.

Total Surface Area

a^2 + 2a\sqrt{\left(\dfrac{a^2}{4} + h^2\right)}

gives

or

a^2 + 2al

gives

.

Important Notes on Square Pyramid

A square pyramid is a three-dimensional geometric shape that has a square base and four triangular sides that are joined at a vertex.

Jai Ganesh · 2023-02-06 22:25:37

Cone

A cone is a three-dimensional shape that has a circular base and it narrows down to a sharp point called a vertex. One of the easiest real-life examples that could be given is a birthday cap in the shape of a cone. With regards to a cone, we have two types of areas. One is the total surface area and the other is a curved surface area. The total surface area of a cone is defined as the area covered by its base and the curved part of the cone, whereas the curved surface area is defined as the area of the curved surfaces of the cone only.

Cone Definition

A cone is a three-dimensional solid geometric shape having a circular base and a pointed edge at the top called the apex. A cone has one face and a vertex. There are no edges for a cone.

The three elements of the cone are its radius, height, and slant height. Radius 'r' is defined as the distance between the center of the circular base to any point on the circumference of the base. Height 'h' of the cone is defined as the distance between the apex of the cone to the center of the circular base. The slant height 'l' is defined as the distance between the apex of the cone to any point on the circumference of the cone. Some of the real-life examples of a cone include a birthday cap, a tent, and a road divider.

Properties of Cone

A cone is a shape that has a curved surface and a circular base. The following properties of a cone help us identify it easily. They are as follows.

* A base of a cone is circular.
* There is one face, one vertex, and no edges for a cone.
* The slant height of a cone is the length of the line segment joining the apex of the cone to any point on the circumference of the base of the cone.
* A cone that has its apex right above the circular base at a perpendicular distance is called a right circular cone.
* A cone that does not have its apex directly above the circular base is an oblique cone.

Cone Formula

There are three important formulas related to a cone. They are the slant height of a cone, the volume of a cone, and its surface area. The slant height of a cone is obtained by finding the sum of the squares of radius and the height of the cylinder which is given by the formula given below.

(slant \ height)^2 = {radius}^2 + {height}^2

gives

If the slant height of the cone is 'l' and the height is 'h' and the radius is 'r', then

l^2 = r^2 + h^2

gives

.

The formula for the slant height of the cone is 'l' =

\sqrt{r^2 + h^2}

gives

.

Curved Surface Area of Cone

The curved surface area of a cone is the area enclosed by the curved part of the cone. For a cone of radius 'r', height 'h', and slant height 'l', the curved surface area is as follows:

Curved \ Surface \ Area = \pi{r}l \ square \ units

gives

.

Total Surface Area of Cone

Total surface area is the sum of the area of the circular base and the area of the curved part of the cone. In other words, it is the sum of the curved surface area of the cone and the area of the circular base, which can be written mathematically as:

Total Surface Area (TSA) = Area of the base (Circle) + Curved Surface Area of the Cone(CSA).

TSA = \pi{r^2} + \pi{r}l \ square \ units

gives

.

Total surface area is sometimes referred to as only surface area. So, whenever we are asked to calculate the surface area of the cone, it means we have to find the total surface area.

Volume of a Cone

Volume = \dfrac{1}{3}\pi{r^2}h \ cubic \ units

gives

.

Let A = Area of base of the cone and h = height of the cone.

Therefore, the volume of cone =

\dfrac{1}{3} \times A \times h

gives

.

Since the base of the cone is circular, we substitute the area to be

\pi{r^2}

gives

.

Volume of cone =

\dfrac{1}{3}\pi{r^2}h \ cubic \ units

gives

.

Also, the volume of a cone is one-third of the volume of a cylinder.

Volume of cone = (1/3) × volume of a cylinder.

Types of Cone

Broadly there are two types of Cones. One is the right circular cone and the other is an oblique cone.

Right Circular Cone : Oblique Cone

A right circular cone has its vertex opposite to the circular base.
An oblique cone does not have its vertex directly opposite to the circular base.

The line representing the height of the cone passes through the center of the base circle and is perpendicular to the radius.
The line representing the height of the cone does not pass through the center of the base circle.

Jai Ganesh · 2023-02-15 01:43:27

Parallelepiped

In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term rhomboid is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, the four concepts—parallelepiped and cube in three dimensions, parallelogram and square in two dimensions—are defined, but in the context of a more general affine geometry, in which angles are not differentiated, only parallelograms and parallelepipeds exist. Three equivalent definitions of parallelepiped are

* a polyhedron with six faces (hexahedron), each of which is a parallelogram,
* a hexahedron with three pairs of parallel faces, and
* a prism of which the base is a parallelogram.

The rectangular cuboid (six rectangular faces), cube (six square faces), and the rhombohedron (six rhombus faces) are all specific cases of parallelepiped.

Parallelepipeds are a subclass of the prismatoids.

A parallelepiped is a three-dimensional shape that is formed by six parallelograms. The word 'parallelepiped' is derived from the Greek word parallelepipdon, meaning "a body having parallel bodies". We can say that a parallelepiped relates with a parallelogram just like a cube relates with a square. Parallelepiped has 6 parallelogram-shaped faces, 8 vertices, and 12 edges. Let us understand properties and different formulas associated with a surface area and volume of a parallelepiped in the following sections.

What Is a Parallelepiped?

A parallelepiped is a three-dimensional shape with six faces, that are all in the shape of a parallelogram. It has 6 faces, 8 vertices, and 12 edges. Cube, cuboid, and rhomboid are all special cases of a parallelepiped. A cube is a parallelepiped whose all sides are square-shaped. Similarly, a cuboid and a rhomboid are parallelepipeds with rectangle and rhombus-shaped faces respectively. In the figure given below, we can observe a parallelepiped, with 'a', 'b', and 'c' as side lengths and 'h' as the height of the parallelepiped.

Properties of Parallelepiped

There are certain properties of a parallelepiped that help us distinguish it from other 3-D shapes. These properties are listed below,

* Parallelepiped is a three-dimensional solid shape.
* It has 6 faces, 12 edges, and 8 vertices.
* All faces of a parallelepiped are in the shape of a parallelogram.
* A parallelepiped has 2 diagonals on each face, called the face diagonals. It has a total of 12 face diagonals.
* The diagonals connecting the vertices not lying on the same face are called the body or space diagonal of a parallelepiped.
* Parallelepiped is referred to as a prism with a parallelogram-shaped base.
* Each face of a parallelepiped is a mirror image of the opposite face.

Surface Area of Parallelepiped

The surface area of a parallelepiped is defined as the total area covered by all the surfaces of a parallelepiped. The surface area of a parallelepiped is expressed in square units, like

{in}^2, {cm}^2, m^2, {ft}^2, {yd}^2

gives

, etc. The surface area of parallelepiped can be of two types:

* Lateral Surface Area
* Total Surface Area

Lateral Surface Area of Parallelepiped

The lateral surface area of a parallelepiped is defined as the area of the lateral or side faces of a parallelepiped. To calculate the LSA of a parallelepiped, we need to find the sum of the area covered by the 4 side faces.

Total Surface Area of Parallelepiped

The total surface area of a parallelepiped is defined as the area of all the faces of a parallelepiped. To calculate the TSA of a parallelepiped, we need to find the sum of the area covered by the 6 faces.

Surface Area of Parallelepiped Formula

The formula to calculate the lateral surface area and total surface area of parallelepiped is given as,

LSA \ of \ Parallelepiped = P \times H

gives

TSA \ of \ Parallelepiped = LSA + 2 \times B = (P \times H) + (2 \times B)

gives

where,

B = Base area
H = Height of parallelepiped
P = Perimeter of base

Volume of Parallelepiped

The volume of a parallelepiped is defined as the space occupied by the shape in a three-dimensional plane. The volume of a parallelepiped is expressed in cubic units, like

{in}^3, {cm}^3, {m}^3, {ft}^3, {yd}^3, \ etc

gives

.

Volume of Parallelepiped Formula

Volume of parallelepiped can be calculated using the base area and the height. The formula to calculate the volume of a parallelepiped is given as,

V = B \times H

gives

.
where,

B = Base area
H = Height of parallelepiped.

Jai Ganesh · 2023-02-19 22:39:29

Parabola

In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

One description of a parabola involves a point (the focus) and a line (the directrix). The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane parallel to another plane that is tangential to the conical surface.[a]

The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the "axis of symmetry". The point where the parabola intersects its axis of symmetry is called the "vertex" and is the point where the parabola is most sharply curved. The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". The "latus rectum" is the chord of the parabola that is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola—that is, all parabolas are geometrically similar.

Parabolas have the property that, if they are made of material that reflects light, then light that travels parallel to the axis of symmetry of a parabola and strikes its concave side is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected into a parallel ("collimated") beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other waves. This reflective property is the basis of many practical uses of parabolas.

The parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors and the design of ballistic missiles. It is frequently used in physics, engineering, and many other areas.

Parabola

A parabola is a graph of a quadratic function. Pascal stated that a parabola is a projection of a circle. Galileo explained that projectiles falling under the effect of uniform gravity follow a path called a parabolic path. Many physical motions of bodies follow a curvilinear path which is in the shape of a parabola. In mathematics, any plane curve which is mirror-symmetrical and usually is of approximately U shape is called a parabola. Here we shall aim at understanding the derivation of the standard formula of a parabola, the different standard forms of a parabola, and the properties of a parabola.

What is Parabola?

A parabola refers to an equation of a curve, such that a point on the curve is equidistant from a fixed point, and a fixed line. The fixed point is called the focus of the parabola, and the fixed line is called the directrix of the parabola. Also, an important point to note is that the fixed point does not lie on the fixed line. A locus of any point which is equidistant from a given point (focus) and a given line (directrix) is called a parabola. Parabola is an important curve of the conic sections of the coordinate geometry.

Parabola Equation

The general equation of a parabola is:

y = a(x-h)^2 + k \ or \ x = a(y-k)^2 +h

gives

, where (h,k) denotes the vertex.

The standard equation of a regular parabola is

y^2 = 4ax

gives

.

Some of the important terms below are helpful to understand the features and parts of a parabola.

* Focus: The point (a, 0) is the focus of the parabola
* Directrix: The line drawn parallel to the y-axis and passing through the point (-a, 0) is the directrix of the parabola. The directrix is perpendicular to the axis of the parabola.
* Focal Chord: The focal chord of a parabola is the chord passing through the focus of the parabola. The focal chord cuts the parabola at two distinct points.
* Focal Distance: The distance of a point

(x_1,y_1)

gives

on the parabola, from the focus, is the focal distance. The focal distance is also equal to the perpendicular distance of this point from the directrix.
* Latus Rectum: It is the focal chord that is perpendicular to the axis of the parabola and is passing through the focus of the parabola. The length of the latus rectum is taken as LL' = 4a. The endpoints of the latus rectum are (a, 2a), (a, -2a).
Eccentricity: (e = 1). It is the ratio of the distance of a point from the focus, to the distance of the point from the directrix. The eccentricity of a parabola is equal to 1.

Standard Equations of a Parabola

There are four standard equations of a parabola. The four standard forms are based on the axis and the orientation of the parabola. The transverse axis and the conjugate axis of each of these parabolas are different. The below image presents the four standard equations and forms of the parabola.

The following are the observations made from the standard form of equations:

* Parabola is symmetric with respect to its axis. If the equation has the term with

y^2

gives

, then the axis of symmetry is along the x-axis and if the equation has the term with

x^2

gives

, then the axis of symmetry is along the y-axis.
* When the axis of symmetry is along the x-axis, the parabola opens to the right if the coefficient of the x is positive and opens to the left if the coefficient of x is negative.
* When the axis of symmetry is along the y-axis, the parabola opens upwards if the coefficient of y is positive and opens downwards if the coefficient of y is negative.

Parabola Formula

Parabola Formula helps in representing the general form of the parabolic path in the plane. The following are the formulas that are used to get the parameters of a parabola.

* The direction of the parabola is determined by the value of a.
* Vertex = (h,k), where h = -b/2a and k = f(h)
* Latus Rectum = 4a
* Focus: (h, k+ (1/4a))
* Directrix: y = k - 1/4a

Derivation of Parabola Equation

Let us consider a point P with coordinates (x, y) on the parabola. As per the definition of a parabola, the distance of this point from the focus F is equal to the distance of this point P from the Directrix. Here we consider a point B on the directrix, and the perpendicular distance PB is taken for calculations.

As per this definition of the eccentricity of the parabola, we have PF = PB (Since e = PF/PB = 1)

The coordinates of the focus is F(a,0) and we can use the coordinate distance formula to find its distance from P(x, y)

PF = \sqrt{(x - a)^2 + (y - 0)^2} = \sqrt{(x - a)^2 + y^2}

gives

The equation of the directrix is x + a = 0 and we use the perpendicular distance formula to find PB.

Squaring the equation on both sides,

(x - a)^2 + y^2 = (x + a)^2

gives

x^2 + a^2 - 2ax + y^2 = x^2 + a^2 + 2ax

gives

y^2 - 2ax = 2ax

gives

y^2 = 4ax

gives

Jai Ganesh · 2023-03-01 15:43:04

Ellipse

In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same.

{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1.}

gives

Ellipse

Ellipse is an integral part of the conic section and is similar in properties to a circle. Unlike the circle, an ellipse is oval in shape. An ellipse has an eccentricity less than one, and it represents the locus of points, the sum of whose distances from the two foci of the ellipse is a constant value. A simple example of the ellipse in our daily life is the shape of an egg in a two-dimensional form and the running tracking in a sports stadium.

Here we shall aim at knowing the definition of an ellipse, the derivation of the equation of an ellipse, and the different standard forms of equations of the ellipse.

What is an Ellipse?

An ellipse in math is the locus of points in a plane in such a way that their distance from a fixed point has a constant ratio of 'e' to its distance from a fixed line (less than 1). The ellipse is a part of the conic section, which is the intersection of a cone with a plane that does not intersect the cone's base. The fixed point is called the focus and is denoted by S, the constant ratio 'e' as the eccentricity, and the fixed line is called as directrix (d) of the ellipse.

Ellipse Definition

An ellipse is the locus of points in a plane, the sum of whose distances from two fixed points is a constant value. The two fixed points are called the foci of the ellipse.

The general equation of an ellipse is used to algebraically represent an ellipse in the coordinate plane. The equation of an ellipse can be given as,

\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1.

gives

Parts of an Ellipse

Let us go through a few important terms relating to different parts of an ellipse.

* Focus: The ellipse has two foci and their coordinates are F(c, o), and F'(-c, 0). The distance between the foci is thus equal to 2c.
* Center: The midpoint of the line joining the two foci is called the center of the ellipse.
* Major Axis: The length of the major axis of the ellipse is 2a units, and the end vertices of this major axis is (a, 0), (-a, 0) respectively.
* Minor Axis: The length of the minor axis of the ellipse is 2b units and the end vertices of the minor axis is (0, b), and (0, -b) respectively.
* Latus Rectum: The latus rectum is a line drawn perpendicular to the transverse axis of the ellipse and is passing through the foci of the ellipse. The length of the latus rectum of the ellipse is

{2b^2}/a

gives

.
* Transverse Axis: The line passing through the two foci and the center of the ellipse is called the transverse axis.
* Conjugate Axis: The line passing through the center of the ellipse and perpendicular to the transverse axis is called the conjugate axis
* Eccentricity: (e < 1). The ratio of the distance of the focus from the center of the ellipse, and the distance of one end of the ellipse from the center of the ellipse. If the distance of the focus from the center of the ellipse is 'c' and the distance of the end of the ellipse from the center is 'a', then eccentricity e = c/a.

Standard Equation of an Ellipse

There are two standard equations of the ellipse. These equations are based on the transverse axis and the conjugate axis of each of the ellipse. The standard equation of the ellipse

\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1

gives

has the transverse axis as the x-axis and the conjugate axis as the y-axis. Further, another standard equation of the ellipse is

\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1

gives

and it has the transverse axis as the y-axis and its conjugate axis as the x-axis. The below image shows the two standard forms of equations of an ellipse.

Area of Ellipse Formula

The area of an ellipse is defined as the total area or region covered by the ellipse in two dimensions and is expressed in square units like

{in}^2, {cm}^2, {m}^2, {yd}^2, {ft}^2,

gives

etc. The area of an ellipse can be calculated with the help of a general formula, given the lengths of the major and minor axis. The area of ellipse formula can be given as,

Area of ellipse =

\pi{a}{b}

gives

where,

a = length of semi-major axis
b = length of semi-minor axis

Eccentricity of an Ellipse Formula

Eccentricity of an ellipse is given as the ratio of the distance of the focus from the center of the ellipse, and the distance of one end of the ellipse from the center of the ellipse

Eccentricity of an ellipse formula,

e = \dfrac{c}{a} = \sqrt{1 - \dfrac{b^2}{a^2}}

gives

Latus Rectum of Ellipse Formula

Latus rectum of of an ellipse can be defined as the line drawn perpendicular to the transverse axis of the ellipse and is passing through the foci of the ellipse. The formula to find the length of latus rectum of an ellipse can be given as,

L = {2b^2}/a

gives

Formula for Equation of an Ellipse

The equation of an ellipse formula helps in representing an ellipse in the algebraic form. The formula to find the equation of an ellipse can be given as,

Equation of the ellipse with centre at (0,0) :

x^2/a^2 + y^2/b^2 = 1

gives

Equation of the ellipse with centre at (h,k) :

(x-h)^2 /a^2 + (y-k)^2/ b^2 =1

gives

.

coolboyx12 · 2023-03-17 04:23:59

Jai Ganesh · 2023-03-17 13:50:09

Annulus

An annulus is an inner region between two concentric circles i.e. two or more circles sharing the same center point. The annulus is shaped like a ring and has many applications in mathematics that we will be learning in this article. Some of the real-life examples are a doughnut, finger rings. etc. Let us learn more about the shape of the annulus and solve a few examples to understand the concept better.

Annulus Definition

An annulus is a two-dimensional flat figure shaped in a circular form which is constructed by two concentric circles. The region or the area formed in between these two concentric circles is called the annulus. Since it is a flat figure in a circular form, the edges are two circles with the same center. It is considered a circular disk having a circular hole in the middle.

Annulus Meaning

The word annulus is derived from a Latin word, 'annuli', meaning little rings. The shape of the annulus is flat and circular with a hole in between, much like a throw ring or a circular disc. Look at the image below showing two circles i.e. one small circle also called an inner circle and a big circle also called the outer circle. The point which is marked as red is the center of both circles. The shaded colored area, between the boundary of these two circles, is known as an annulus.

Area of the Annulus

The annulus area is the area of the ring-shaped space i.e. the enclosed region between the two concentric circles. To calculate the area of the annulus, we need the area of both the inner circle and the outer circle. The dimensions of an annulus are defined by the two radii R, and r, which are the radii of the outer ring and the inner ring respectively. Once the measurements of both the radii are known, we can calculate the area by subtracting the area of the small circle from the big circle. Hence, the formula used for finding the area of the annulus is:

Area of Outer Circle =

\pi{R}^2

gives

.

Area of Inner Circle =

\pi{r}^2

gives

.

Area of Annulus = Area of Outer Circle – Area of Inner Circle

Therefore, Area of Annulus =

\pi(R^2 - r^2)

gives

square units, or it can be written as

\pi(R + r)(R - r)

gives

square units, where R is the radius of the outer circle, r is the radius of the inner circle, and π(pi) is approximately 3.142.

The area of the outer (bigger) circle - the area of the inner (smaller) circle = the area of the annulus.

Annulus Perimeter

The perimeter is the distance around the 2D shape. Since the annulus is a flat circular shape constructed by two concentric circles, it can also be considered as a ring. Therefore, an open ring can be considered as the topological equivalent of a cylinder and a punctured plane. Similar to the area, to find the perimeter of the annulus we need to consider both the inner circle and the outer circle. So, the perimeter of the ring or annulus is equal to the sum of the radii of the large and small circles multiplied by 2π. The formula for finding the perimeter is:

Perimeter of Annulus (P) =

2\pi(R + r)

gives

units, where R is the radius of the outer circle, r is the radius of the inner circle, and

\pi

gives

(pi) is approximately 3.142.

Jai Ganesh · 2023-03-27 17:58:13

Hyperbola

In mathematics, a hyperbola is an important conic section formed by the intersection of the double cone by a plane surface, but not necessarily at the center. A hyperbola is symmetric along the conjugate axis, and shares many similarities with the ellipse. Concepts like foci, directrix, latus rectum, eccentricity, apply to a hyperbola. A few common examples of hyperbola include the path followed by the tip of the shadow of a sundial, the scattering trajectory of sub-atomic particles, etc.

Here we shall aim at understanding the definition, formula of a hyperbola, derivation of the formula, and standard forms of hyperbola using the solved examples.

What is Hyperbola?

A hyperbola, a type of smooth curve lying in a plane, has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. A hyperbola is a set of points whose difference of distances from two foci is a constant value. This difference is taken from the distance from the farther focus and then the distance from the nearer focus. For a point P(x, y) on the hyperbola and for two foci F, F', the locus of the hyperbola is PF - PF' = 2a.

Hyperbola Definition

A hyperbola, in analytic geometry, is a conic section that is formed when a plane intersects a double right circular cone at an angle such that both halves of the cone are intersected. This intersection of the plane and cone produces two separate unbounded curves that are mirror images of each other called a hyperbola.

Parts of a Hyperbola

Let us check through a few important terms relating to the different parameters of a hyperbola.

Foci of hyperbola: The hyperbola has two foci and their coordinates are F(c, o), and F'(-c, 0).

Center of Hyperbola: The midpoint of the line joining the two foci is called the center of the hyperbola.

Major Axis: The length of the major axis of the hyperbola is 2a units.

Minor Axis: The length of the minor axis of the hyperbola is 2b units.

Vertices: The points where the hyperbola intersects the axis are called the vertices. The vertices of the hyperbola are (a, 0), (-a, 0).

Latus Rectum of Hyperbola: The latus rectum is a line drawn perpendicular to the transverse axis of the hyperbola and is passing through the foci of the hyperbola. The length of the latus rectum of the hyperbola is

2b^2/a

gives

.

Transverse Axis: The line passing through the two foci and the center of the hyperbola is called the transverse axis of the hyperbola.

Conjugate Axis: The line passing through the center of the hyperbola and perpendicular to the transverse axis is called the conjugate axis of the hyperbola.

Eccentricity of Hyperbola: (e > 1) The eccentricity is the ratio of the distance of the focus from the center of the hyperbola, and the distance of the vertex from the center of the hyperbola. The distance of the focus is 'c' units, and the distance of the vertex is 'a' units, and hence the eccentricity is e = c/a.

Hyperbola Equation

The below equation represents the general equation of a hyperbola. Here the x-axis is the transverse axis of the hyperbola, and the y-axis is the conjugate axis of the hyperbola.

\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1

gives

.

Let us understand the standard form of the hyperbola equation and its derivation in detail in the following sections.

Standard Equation of Hyperbola

There are two standard equations of the Hyperbola. These equations are based on the transverse axis and the conjugate axis of each of the hyperbola. The standard equation of the hyperbola is

\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1

gives

.

has the transverse axis as the x-axis and the conjugate axis is the y-axis. Further, another standard equation of the hyperbola is

\dfrac{y^2}{a^2} - \drac{x^2}{b^2} = 1

gives

and it has the transverse axis as the y-axis and its conjugate axis is the x-axis.

Eccentricity(e) of hyperbola formula:

e  = \sqrt{1 + \dfrac{b^2}{a^2}}

gives

.

Dkor · 2023-05-01 19:21:11

this really helped.

Jai Ganesh · 2023-05-01 20:03:06

Rectangular Hyperbola

Rectangular Hyperbola is a hyperbola having the transverse axis and the conjugate of 2a units and conjugate axis of 2b units of equal length. The eccentricity of a rectangular hyperbola is

\sqrt{2}

gives

, and the equation of a rectangular hyperbola is

x^2 - y^2 = a^2

gives

What Is A Rectangular Hyperbola?

A rectangular hyperbola is a hyperbola having the transverse axis and the conjugate axis of equal length. The arcs of a rectangular hyperbola is the same as the arc of a circle. For a rectangular hyperbola having the transverse axis of length 2a and the conjugate axis of length 2b, we have 2a = 2b, or a = b. The general equation of a rectangular hyperbola is

x^2 - y^2 = a^2.

gives

The equation of asymptotes of a rectangular hyperbola is

y = \pm \ or \ x \ or \ x^2 - y^2 = 0

gives

The axes or the asymptotes of the rectangular hyperbola are perpendicular to each other. The rectangular hyperbola is related to a hyperbola in a similar form as the circle is related to an ellipse. The eccentricity of a rectangular hyperbola is

\sqrt{2}

gives

The graph of the equation y = 1/x is similar to the graph of a rectangular hyperbola.

Properties of Rectangular Hyperbola

The rectangular hyperbola is similar to a regular hyperbola, and the only difference is the different lengths of the transverse axis and conjugate axis in a hyperbola, and these lengths are equal in a rectangular hyperbola The following are some of the important properties of a rectangular hyperbola.

* The eccentricity of a rectangular hyperbola is equal to

\sqrt{2}

gives

* The transverse axis and the conjugate axis in a rectangular hyperbola is of equal length.

* The asymptotoes of a rectangular hyperbola is

y = \pm \ x \ or \ x^2 - y^2 = 0

gives

* The asymptotes of a rectangular hyperbola are perpendicular to each other.

* The conjugate of a rectangular hyperbola

x^2 - y^2 = a^2

gives

is also a rectangular hyperbola

x^2 - y^2 = -a^2

gives

* The parametric form of representation of a rectangular hyperbola has the coordinates

x = aSec\theta, y = aTan\theta

gives

Jai Ganesh · 2023-08-18 20:28:02

Commutative Property

The commutative property applies to the arithmetic operations of addition and multiplication. It means that changing the order or position of two numbers while adding or multiplying them does not change the end result. For example, 4 + 5 gives 9, and 5 + 4 also gives 9. The order of two numbers being added does not affect the sum. The same concept applies to multiplication too. The commutative property does not hold for subtraction and division, as the end results are completely different after changing the order of numbers.

What is Commutative Property?

The word 'commutative' originates from the word 'commute', which means to move around. Hence, the commutative property deals with moving the numbers around. So mathematically, if changing the order of the operands does not change the result of the arithmetic operation then that particular arithmetic operation is commutative. Let us discuss the commutative property of addition and multiplication.

Commutative Property Formula

For any two numbers, A and B, the formula of the commutative property of numbers is expressed as follows.

A + B = B + A

gives

.

A \times B = B \times A

gives

A - B \neq B - A

gives

A \div B \neq B \div A

gives

.

The commutative property formula states that the change in the order of two numbers while adding and multiplying them does not affect the result. However, while subtracting and dividing any two real numbers, the order of numbers are important and hence it can't be changed.

Commutative Property of Addition

The commutative property of addition says that changing the order of the addends does not change the value of the sum. If 'A' and 'B' are two numbers, then the commutative property of addition of numbers can be represented as shown in the figure given below.

Commutative Property of Addition Formula

Let us take an example of the commutative property of addition and understand the application of the above formula.

Example: Let us check the Commutative property by adding 10 and 13.

Let us add the given numbers 10 and 13. So, 10 + 13 = 23 and 13 + 10 = 23. Therefore, 10 + 13 = 13 + 10 which proves the commutative property of addition.

Commutative Property of Multiplication

The commutative property of multiplication says that the order in which we multiply two numbers does not change the final product. The figure given below represents the commutative property of the multiplication of two numbers.

Commutative Property of Multiplication

If 4 and 6 are the numbers, then

4 \times 6 = 24, \ and \ 6 \times 4

gives

is also equal to 24. Thus

4 \times 6 = 6 \times 4

gives

. Therefore, the commutative property holds true for the multiplication of numbers.

Note: The commutative property does not hold for subtraction and division operations. Let us take the example of numbers 6 and 2.

6 - 2 = 4, \ but \ 2 - 6 = -4. \ Thus,\ 6 - 2 \neq 2 - 6

gives

.

6 \div 2 = 3, \ but \ 2 \div 6 = 1/3. \ Thus, 6 \div 2 \neq 2 \div 6

gives

.

Commutative Property of Subtraction

The commutative property is not applicable to subtraction. The commutative law only applies to addition and multiplication. Let us see why it does not apply on subtraction. For example,

8 - 5 = 3, \ but \ 5 - 8 = -3. \ Thus, 8 - 5 \neq 5 - 8

gives

.

Jai Ganesh · 2023-10-01 21:24:02

Associative Property

The associative property, or the associative law in mathematics, states that while adding or multiplying numbers, the way in which numbers are grouped by brackets (parentheses), does not affect their sum or product. The associative property is applicable to addition and multiplication. Let us learn more about the associative property, along with some associative property examples.

What is the Associative Property?

According to the Associative property, when 3 or more numbers are added or multiplied, the result (sum or the product) remains the same even if the numbers are grouped in a different way. Here, grouping is done with the help of brackets. This can be expressed as,

a \times (b \times c) = (a \times b) \times c

gives

and

a + (b + c) = (a + b) + c

gives

.

Associative Property Definition

The associative law which applies only to addition and multiplication states that the sum or the product of any 3 or more numbers is not affected by the way in which the numbers are grouped by parentheses. In other words, if the same numbers are grouped in a different way for addition and multiplication, their result remains the same.

The formula for the associative property of addition and multiplication is expressed as:

Formula for the Associative law of addition and multiplication

Let us discuss in detail the associative property of addition and multiplication with examples.

Associative Property of Addition

According to the associative property of addition, the sum of three or more numbers remains the same irrespective of the way the numbers are grouped. Suppose we have three numbers: a, b, and c. For these, the associative property of addition will be expressed with the following formula:

Associative Property of Addition Formula:

(A + B) + C = A + (B + C)

gives

.

Let us understand this with the help of an example.

Example: (1 + 7) + 3 = 1 + (7 + 3) = 11. If we solve the left-hand side, we get, 8 + 3 = 11. Now, if we solve the right-hand side, we get, 1 + 10 = 11. Hence, we can see that the sum remains the same even when the numbers are grouped in a different way.

Associative Property of Multiplication

The associative property of multiplication states that the product of three or more numbers remains the same irrespective of the way the numbers are grouped. The associative property of multiplication can be expressed with the help of the following formula:

Associative Law of Multiplication Formula

(A \times B) \times C = A \times (B \times C)

gives

.
Let us understand this with the following example.

Example:

(1 \times 7) \times 3 = 1 \times (7 \times 3) = 21

gives

.

When we solve the left-hand side, we get

7 × 3 = 21

gives

Now, when we solve the right-hand side, we get

1 × 21 = 21

gives

.

Therefore, it can be seen that the product of the numbers remains the same irrespective of the different grouping of numbers.

Associative Property of Subtraction

The associative property does not work with subtraction. This means if we try to apply the associative law to subtraction, it will not work. For example, (7 - 1) - 3 is not equal to 7 - (1 - 3). If we solve the left-hand side, we get, 6 - 3 = 3. Now, if we solve the right-hand side, we get, 7 - (-2) = 9. Hence, we can see there is no associative property of subtraction.

Verification of Associative Law

Let us try to justify how and why the associative property is only valid for addition and multiplication operations. We will apply the associative law individually on the four basic operations.

For Addition: The associative law in Maths for addition is expressed as (A + B) + C = A + (B + C). So, let us substitute this formula with numbers to verify it. For example, (1 + 4) + 2 = 1 + (4 + 2) = 7. Therefore, the associative property is applicable to addition.

For Subtraction: Let us try the associative property formula in subtraction. This can be expressed as (A - B) - C ≠ A - (B - C). Now, let us verify this formula by substituting numbers in this. For example,

(1 - 4) - 2 \neq 1 - (4 - 2)

gives

i.e., -5 ≠ -1. Therefore, we say that the associative property is not applicable to subtraction.

For Multiplication: The associative law for multiplication is given as

(A \times B) \times C = A \times (B \times C)

gives

For example,

(1 \times 4) \times 2 = 1 \times (4 \times 2) = 8

gives

Therefore, we can say that the associative property is applicable to multiplication.

For Division: Now, let us try the associative property formula for division. This can be expressed as

(A \div B) \div C \neq A \div (B \div C)

gives

For example,

(9 \div 3) \div 2 \neq 9 \div (3 \div 2) = 3/2 \neq 6

gives

.
Therefore, we can see that the associative property is not applicable to division.

Jai Ganesh · 2023-10-15 22:02:44

Distributive Property

What is the distributive property?

Solution:

The distributive property is a property of multiplication used in addition and subtraction.

Distributive property states that multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number and then adding the products together.

It also applies for difference of numbers being multiplied to a number.

It is given as:

a \times (b + c) = a \times b + a \times c

gives

.

a \times (b - c) = a \times b - a \times c

gives

Example 1 :

3 \times ( 2 + 5 ) = 3 \times 2 + 3 \times 5

gives

LHS = 3 \times ( 2 + 5 ) = 3 \times 7 = 21

gives

RHS = 3 \times 2 + 3 \times 5 = 6 + 15 = 21

gives

Thus, LHS = RHS

Example 2 :

5 \times ( 4 - 1) = 5 \times 4 - 5 \times 1

gives

.

LHS = 5 \times ( 4 - 1) = 5 \times 3 = 15

gives

RHS = 5 \times 4 - 5 \times 1 = 20 - 5 = 15

gives

Hence, LHS = RHS

Thus, we have understood the usage of the distributive property by looking into the examples.

What is the distributive property?

Summary:

Distributive property states that two or more terms in addition or subtraction, when multiplied with a number are equal to the addition or subtraction of the product of each of the terms with that number.

Jai Ganesh · 2023-10-17 16:49:28

Incenter

In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisectors of the triangle cross, as the point equidistant from the triangle's sides, as the junction point of the medial axis and innermost point of the grassfire transform of the triangle, and as the center point of the inscribed circle of the triangle.

Incenter Formula

To calculate the incenter of a triangle with 3 cordinates, we can use the incenter formula. Let us learn about the formula. Consider the coordinates of incenter of the triangle ABC with coordinates of the vertices,

A((x)_1, (y)_1), B((x)_2, (y)_2), C((x)_3, (y)_3)

gives

and sides a, b, c are:

\left(\dfrac{ax_1 + bx_2 + cx_3}{a + b+ c}, \dfrac{ay_1 + by_2 + cy_3}{a + b+ c}\right)

gives

.

amnkb · 2023-11-14 04:34:28

jane posted here: https://www.mathisfunforum.com/viewtopic.php?pid=61290#p61290

JaneFairfax wrote:

You can even specify exactly how much spacing you want. For example, if you want exactly 30 mm of white space,
\hspace{30mm}
Thus:

but text isnt indented
is there a way to indent?
\quad doesnt work
\text{ } doesnt work

p.s. nehushtn posted here: https://www.mathisfunforum.com/viewtopic.php?pid=291982#p291982

Nehushtan wrote:

that is so frikkin awesome!

Jai Ganesh · 2023-11-14 20:57:57

To indent subsequent lines of a paragraph, use the TeX command \hangindent . (While the default behaviour is to apply the hanging indent after the first line, this may be changed with the \hangafter command.) An example follows. \hangindent=0.7cm This paragraph has an extra indentation at the left.

This link helps:

LaTeX/Paragraph Formatting.

amnkb · 2023-11-14 22:59:26

Jai Ganesh wrote:

To indent subsequent lines of a paragraph, use the TeX command \hangindent . (While the default behaviour is to apply the hanging indent after the first line, this may be changed with the \hangafter command.) An example follows. \hangindent=0.7cm This paragraph has an extra indentation at the left.
This link helps:
LaTeX/Paragraph Formatting.

thanks!

Jai Ganesh · 2023-11-15 14:14:16

Centroid of Triangle (Centroid)

The centroid of a triangle is a point of concurrency of the medians of a triangle. Before understanding the point of concurrency, let us discuss the medians of a triangle. Medians are the line segments that are drawn from the vertex to the mid-point of the opposite side of the vertex. Each median of a triangle divides the triangle into two smaller triangles that have equal areas. The point of intersection of the medians of a triangle is known as centroid. The centroid always lies inside a triangle, unlike other points of concurrencies of a triangle.

What is Centroid of a Triangle?

The centroid of a triangle is formed when three medians of a triangle intersect. It is one of the four points of concurrencies of a triangle. The medians of a triangle are constructed when the vertices of a triangle are joined with the midpoint of the opposite sides of the triangle. ntroid of a triangle

Properties of the Centroid of Triangle

The following points show the properties of the centroid of a triangle which are very helpful to distinguish the centroid from all the other points of concurrencies.

* The centroid is also known as the geometric center of the object.
* The centroid of a triangle is the point of intersection of all the three medians of a triangle.
* The medians are divided into a 2:1 ratio by the centroid.
* The centroid of a triangle is always within a triangle.

Centroid of Triangle Formula

The centroid of a triangle formula is used to find the centroid of a triangle uses the coordinates of the vertices of a triangle. The coordinates of the centroid of a triangle can only be calculated if we know the coordinates of the vertices of the triangle. The formula for the centroid of the triangle is:

C(x,y) = (x_1 + x_2 + x_3)/3, (y_1 + y_2 + y_3)/3

gives

or

C(x,y) = \left(\dfrac{x_1 + x_2 + x_3}{3}, \dfrac{y_1 + y_2 + y_3}{3}\right)

gives

where, x1, x2, and x3 are the 'x-coordinates' of the vertices of the triangle; and y1, y2, and y3 are the 'y-coordinates of the vertices of the triangle.

Important Notes on Centroid of Triangle

* The centroid of a triangle is the point of intersection of the medians of a triangle.
* It always lies inside the triangle.
* Centroid divides the medians in the ratio 2:1.

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#226 2023-02-04 17:45:36

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