Math Is Fun Forum

  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#1 2014-03-27 17:25:19

niharika_kumar
Member
From: Numeraland
Registered: 2013-02-12
Posts: 1,062

sets

If

  and
,
then
contains

a) one point
b) three points
c) two points
d) four points

P.S. I am not able to get the question clearly, i.e, what is exactly asked here. Pls. explain the question.

Last edited by niharika_kumar (2014-03-27 17:25:52)


friendship is tan 90°.

Offline

#2 2014-03-27 18:01:12

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: sets

It says A is the set of all ordered pairs of x and y that fits the given equation and so is B.

How many points do A and B have in common?

You have started studying for XI?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

Offline

#3 2014-03-27 18:06:27

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: sets

I am getting four intersection points.


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

Offline

#4 2014-03-27 20:26:20

Bob
Administrator
Registered: 2010-06-20
Posts: 10,058

Re: sets

hi niharika

The first is the equation for a circle and the second, an ellipse.  Both 'centred' on the origin, so, by symmetry, if they intersect at all, there must be 4 points of intersection.  Eliminate x squared by subtracting and you can make y the subject and get two values.

Further thought:  The minor axis of the ellipse is from (0,-4) to (0,+4) and the radius of the circle is 5, so you can tell that they intersect without any algebra.  smile

Bob

Last edited by Bob (2014-03-27 20:29:49)


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 smile

Offline

#5 2014-03-27 20:37:19

niharika_kumar
Member
From: Numeraland
Registered: 2013-02-12
Posts: 1,062

Re: sets

and what if i want to solve it using algebra. We must have that handy too and I am not able to solve that using that sad




agnishom wrote:

You have started studying for XI?

yes i have started a bit from a couple of days.It seems you have covered a lot of portion.

Last edited by niharika_kumar (2014-03-27 20:37:47)


friendship is tan 90°.

Offline

#6 2014-03-27 20:58:48

Bob
Administrator
Registered: 2010-06-20
Posts: 10,058

Re: sets

If

  and
,

Subtracting gives

You don't need to solve that to see it has solutions.

Depending on the relative sizes of the two shapes, you could have a situation where one fits completely inside the other without any intersection.

Bob


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 smile

Offline

#7 2014-03-27 23:54:40

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: sets

agnishom wrote:

You have started studying for XI?

yes i have started a bit from a couple of days.It seems you have covered a lot of portion.

I rarely study seriously. It only so happened that I eventually knew this.

Thought: Set theory is not a theory. Its a language, rather. AS you can see, this problem tests your knowledge of algebra.

The same question put in a different way is: How many ordered pairs of x and y satisfy the system of equations?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

Offline

#8 2014-03-28 10:02:20

eigenguy
Member
Registered: 2014-03-18
Posts: 78

Re: sets

bob bundy wrote:

if they intersect at all, there must be 4 points of intersection.

Or 2 points, but that requires that the circle radius exactly match one of the ellipse semi-axes, which is not the case here.

But, niharika_kumar, I doubt you were intended to solve this problem algebraicly. If you have already studied the equations of circles and ellipses, then it is likely the test preparer hoped for you to recognize them, and then be able to do exactly what Bob did in his first answer. Realize that the circle and ellipse had to intersect in either 0, 2, or 4 points. Then by examination of the radii and axes to realize it had to be 4.

But he or she also made it simple enough to solve for x[sup]2[/sup] and y[sup]2[/sup], and to notice that neither one is negative, meaning that two real values of x and of y exist that solve them, from which 2x2 = 4 points can be constructed.

It is not necessary, or even desirable, to actually figure out exactly what x and y are to solve this problem.

Last edited by eigenguy (2014-03-28 10:11:59)


"Having thus refreshed ourselves in the oasis of a proof, we now turn again into the desert of definitions." - Bröcker & Jänich

Offline

#9 2014-03-28 17:16:49

niharika_kumar
Member
From: Numeraland
Registered: 2013-02-12
Posts: 1,062

Re: sets

i have not yet studied the equations of circles and ellipses but will do it in a short time.
thanks for clarifying my doubt.


friendship is tan 90°.

Offline

Board footer

Powered by FluxBB