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In the xy-plane, point (r, s) lies on a circle with center

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In the xy-plane, point (r, s) lies on a circle with center [#permalink] New post 12 Apr 2005, 13:56
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In the xy-plane, point (r, s) lies on a circle with center at the origin. What is the value of r^2 + s^2?

(1) The circle has radius 2
(2) The point \((\sqrt{2}, \ -\sqrt{2})\) lies on the circle
[Reveal] Spoiler: OA

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Last edited by Bunuel on 11 Feb 2012, 05:32, edited 2 times in total.
Edited the question and added the OA
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Re: DS circle [#permalink] New post 12 Apr 2005, 15:26
saurya_s wrote:
In the xy-plane, point (r, s) lies on a circle with center at the origin. What is the value of r^2 + s^2?
(1) The circle has radius 2.
(2) The point (v2, -v2) lies on the circle.


(1) r^2 + s^2 is the square of the radius of the circle. Sufficient.

(2) This is of no consequence since for any circle centered at the origin, there would be a point (v2. -v2) would lie on the circle. Gives us no info about r^2 + s^2.

Therefore, (A).
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In the xy-plane, point (r, s) lies on a circle with center [#permalink] New post 11 Feb 2012, 05:30
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DeeptiM wrote:
OA is D...can anyone explain??


THEORY:
In an xy-plane, the circle with center (a, b) and radius r is the set of all points (x, y) such that:
\((x-a)^2+(y-b)^2=r^2\)


Image

This equation of the circle follows from the Pythagorean theorem applied to any point on the circle: as shown in the diagram above, the radius is the hypotenuse of a right-angled triangle whose other sides are of length x-a and y-b.

If the circle is centered at the origin (0, 0), then the equation simplifies to: \(x^2+y^2=r^2\).

For more on this subject check Coordinate Geometry chapter of Math Book: math-coordinate-geometry-87652.html

BACK TO THE ORIGINAL QUESTION:
In the xy-plane, point (r, s) lies on a circle with center at the origin. What is the value of \(r^2 + s^2\)?

Now, as \(x^2+y^2=radius^2\) then the question asks about the value of radius^2.

(1) The circle has radius 2 --> radius^2=4. Sufficient.

(2) The point \((\sqrt{2}, \ -\sqrt{2})\) lies on the circle --> substitute x and y coordinates of a point in \(x^2+y^2=radius^2\) --> \(2+2=4=r^2\). Sufficient.

Answer: D.

Hope it helps.
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Re: In the xy-plane, point (r, s) lies on a circle with center [#permalink] New post 02 Mar 2013, 21:08
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Thanks for the brilliant explanation. One thing I don't get the question is that, the point (r,s) could be anywhere in the circle, not only on its circumference. Why does it refer only to a point on the circumference? Thanks!
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Re: In the xy-plane, point (r, s) lies on a circle with center [#permalink] New post 02 Mar 2013, 23:02
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ryusei1989 wrote:
Thanks for the brilliant explanation. One thing I don't get the question is that, the point (r,s) could be anywhere in the circle, not only on its circumference. Why does it refer only to a point on the circumference? Thanks!


It is the language.
On the circle = On the circumference.
In/Inside/Within the circle = Points enclosed by the circumference
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Re: In the xy-plane, point (r, s) lies on a circle with center [#permalink] New post 05 Jul 2014, 01:53
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In the xy-plane, point (r, s) lies on a circle with center [#permalink] New post 09 Jul 2014, 06:28
ryusei1989 wrote:
One thing I don't get the question is that, the point (r,s) could be anywhere in the circle, not only on its circumference.


Exactly.
In my opinion it's just poorly formulated as I've seen this exact questioning angle being used as a trap.
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Re: In the xy-plane, point (r, s) lies on a circle with center [#permalink] New post 09 Jul 2014, 06:30
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mnlsrv wrote:
ryusei1989 wrote:
One thing I don't get the question is that, the point (r,s) could be anywhere in the circle, not only on its circumference.


Exactly.
In my opinion it's just poorly formulated as I've seen this exact questioning angle being used as a trap.


This doubt is addressed here: in-the-xy-plane-point-r-s-lies-on-a-circle-with-center-15566.html#p1191069

Point (r, s) lies ON a circle, means it's on the circumference.
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Re: In the xy-plane, point (r, s) lies on a circle with center [#permalink] New post 09 Jul 2014, 23:52
It is very interesting and difficult question. Thanks for all the solution of this...
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In the xy-plane, point (r, s) lies on a circle with center [#permalink] New post 09 Jul 2015, 01:18
Hi,

i don't understand from Bunuels solution how we come up with the value of S in statement 1 in order to answer what r^2 + s^2 is?

r^2 equals 4, that is all clear, but s should be y-value, how do we get that?
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Re: In the xy-plane, point (r, s) lies on a circle with center [#permalink] New post 09 Jul 2015, 02:43
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noTh1ng wrote:
Hi,

i don't understand from Bunuels solution how we come up with the value of S in statement 1 in order to answer what r^2 + s^2 is?

r^2 equals 4, that is all clear, but s should be y-value, how do we get that?


\(r^2+s^2=radius^2\). (1) says that radius = 2, thus \(r^2+s^2=2^2\).
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RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis ; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) ; 12. Tricky questions from previous years.

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: In the xy-plane, point (r, s) lies on a circle with center [#permalink] New post 09 Jul 2015, 02:52
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noTh1ng wrote:
Hi,

i don't understand from Bunuels solution how we come up with the value of S in statement 1 in order to answer what r^2 + s^2 is?

r^2 equals 4, that is all clear, but s should be y-value, how do we get that?


Hi noTh1ng,

In the xy-plane, point (r, s) lies on a circle with center at the origin. What is the value of r^2 + s^2?

(1) The circle has radius 2
(2) The point (2√, −2√) lies on the circle

You seem to have misunderstood a little here.

The equation of Circle is given by \(x^2 + y^2 = Radius^2\)

Given : (r,s) lie on the circle
i.e. (r,s) will satisfy the equation of Circle
i.e. \(r^2 + s^2 = Radius^2\)

Question : Find the value of \(r^2 + s^2\)? but since \(r^2 + s^2 = Radius^2\) therefore, the question becomes

Question : Find the value of \(Radius^2\)?

Statement 1: The circle has radius 2
i.e. \(r^2 + s^2 = Radius^2 = 2^2 = 4\)
SUFFICIENT

Statement 2: The point (√2, −√2) lies on the circle
i.e. (√2, −√2) will satisfy the equation of circle
i.e. (√2)^2 + (−√2)^2 = Radius^2
i.e. Radius = 4
hence, \(r^2 + s^2 = Radius^2 = 2^2 = 4\) Hence,
SUFFICIENT

Answer: Option D

I hope it helps!

Please Note: You have been confused r (X-co-ordinate) and r (Radius) as it seems from your question
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Re: In the xy-plane, point (r, s) lies on a circle with center   [#permalink] 09 Jul 2015, 02:52
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