Math Is Fun Forum

mathstudent2000

Member

Registered: 2013-07-26

Posts: 79

geometry problems

1. For some positive real number r, the line x + y = r is tangent to the circle x^2 + y^2 = r. Find r.

2. Find the center of the circle passing through the points (-1,0), (1,0), and (3,1). Express your answer in the form "(a,b)."

3. A line with slope 3 is 2 units away from the origin. Find the area of the triangle formed by this line and the coordinate axes.

4. Find the maximum value of y/x over all real numbers x and y that satisfy (x - 3)^2 + (y - 3)^2 = 6.

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Genius is one percent inspiration and ninety-nine percent perspiration

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: geometry problems

Hi mathstudent2000;

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

mathstudent2000

Member

Registered: 2013-07-26

Posts: 79

Re: geometry problems

For no. 3 its not 20/3, but 6 because i tried it yesterday and got it correct

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Genius is one percent inspiration and ninety-nine percent perspiration

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: geometry problems

Who said it was 20 / 3?

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

mathstudent2000

Member

Registered: 2013-07-26

Posts: 79

Re: geometry problems

sorry, i think i saw another problem's answer

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Genius is one percent inspiration and ninety-nine percent perspiration

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: geometry problems

Hi;

We are all done here?

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

champion999

Member

Registered: 2015-07-01

Posts: 2

Re: geometry problems

The correct answer for (3) was actually 20/3.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: geometry problems

Hi;

That is the answer I would have used also but sometimes you have to guess at what the OP wants.

For no. 3 its not 20/3, but 6 because i tried it yesterday and got it correct.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

RandomPieKevin

Member

Registered: 2015-07-02

Posts: 29

Re: geometry problems

Can someone explain how to get the answer for Q4? Thanks.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: geometry problems

I answered this question and now I am wondering how I did it?! What does this x/y mean?

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: geometry problems

You can do it by finding the tangents from (0,0) to the circle.

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“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: geometry problems

You might use Lagrangian Multipliers too.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: geometry problems

4. Find the maximum value of y/x over all real numbers x and y that satisfy (x - 3)^2 + (y - 3)^2 = 6.

That locus is a circle, centre (3,3) and radius root 6.

Pick any point on the circle and join to the origin.  The gradient of the line will be the value of y/x .

To maximise this value move the point around the circle until the line makes a tangent to the circle, at approximately (0.59, 3.38). 

You can calculate the exact point using coordinate geometry.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

math9maniac

Member

From: Tema

Registered: 2015-03-30

Posts: 443

Re: geometry problems

For no. 2, you put the values of x and y into the general equation of the circle, for all three points. You obtain 3 equations with three variables. Solve simultaneously. The center of the circle is (-g, -f).

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Only a friend tells you your face is dirty.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: geometry problems

As already stated by two people you need to make a tangent to the circle from the origin.  One will have the maximum gradient; the other has the minimum.

Let the point where the max tangent hits the circle be P, and the centre of the circle be C, origin O.

Then OC^2 = CP^2 + PO^2 => OP = root 12.

So P is on a circle x^2 + y^2 = 12.

Put this with the other circle to get x + y = 4.  Substitute y = 4 - x into the first circle and solve for x.

You'll get two answers.  One is the max and one the min.

Find y and compute y/x

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob