GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 16 Aug 2018, 20:42

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

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

Author Message
TAGS:

### Hide Tags

Director
Joined: 08 Jul 2004
Posts: 586
In the xy-plane, point (r, s) lies on a circle with center  [#permalink]

### Show Tags

Updated on: 11 Feb 2012, 06:32
2
11
00:00

Difficulty:

25% (medium)

Question Stats:

66% (00:41) correct 34% (00:58) wrong based on 792 sessions

### HideShow timer Statistics

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

_________________

Regards, S

Originally posted by saurya_s on 12 Apr 2005, 14:56.
Last edited by Bunuel on 11 Feb 2012, 06:32, edited 2 times in total.
Edited the question and added the OA
Math Expert
Joined: 02 Sep 2009
Posts: 47946
In the xy-plane, point (r, s) lies on a circle with center  [#permalink]

### Show Tags

11 Feb 2012, 06:30
17
13
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:
$$(x-a)^2+(y-b)^2=r^2$$

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.

(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.

Hope it helps.
_________________
##### General Discussion
Senior Manager
Joined: 15 Mar 2005
Posts: 412
Location: Phoenix

### Show Tags

12 Apr 2005, 16:26
1
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).
_________________

Who says elephants can't dance?

Intern
Joined: 09 Oct 2012
Posts: 1
Re: In the xy-plane, point (r, s) lies on a circle with center  [#permalink]

### Show Tags

02 Mar 2013, 22:08
3
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!
Senior Manager
Joined: 28 Apr 2012
Posts: 298
Location: India
Concentration: Finance, Technology
GMAT 1: 650 Q48 V31
GMAT 2: 770 Q50 V47
WE: Information Technology (Computer Software)
Re: In the xy-plane, point (r, s) lies on a circle with center  [#permalink]

### Show Tags

03 Mar 2013, 00:02
4
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
_________________

"Appreciation is a wonderful thing. It makes what is excellent in others belong to us as well."
― Voltaire

Press Kudos, if I have helped.
Thanks!

Intern
Joined: 29 May 2012
Posts: 2
In the xy-plane, point (r, s) lies on a circle with center  [#permalink]

### Show Tags

09 Jul 2014, 07:28
1
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.
Math Expert
Joined: 02 Sep 2009
Posts: 47946
Re: In the xy-plane, point (r, s) lies on a circle with center  [#permalink]

### Show Tags

09 Jul 2014, 07:30
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.
_________________
Manager
Joined: 07 Apr 2015
Posts: 176
In the xy-plane, point (r, s) lies on a circle with center  [#permalink]

### Show Tags

09 Jul 2015, 02:18
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?
Math Expert
Joined: 02 Sep 2009
Posts: 47946
Re: In the xy-plane, point (r, s) lies on a circle with center  [#permalink]

### Show Tags

09 Jul 2015, 03:43
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 $$r^2+s^2=2^2$$.
_________________
SVP
Joined: 08 Jul 2010
Posts: 2137
Location: India
GMAT: INSIGHT
WE: Education (Education)
Re: In the xy-plane, point (r, s) lies on a circle with center  [#permalink]

### Show Tags

09 Jul 2015, 03:52
2
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
hence, $$r^2 + s^2 = Radius^2 = 2^2 = 4$$ Hence,
SUFFICIENT

I hope it helps!

Please Note: You have been confused r (X-co-ordinate) and r (Radius) as it seems from your question
_________________

Prosper!!!
GMATinsight
Bhoopendra Singh and Dr.Sushma Jha
e-mail: info@GMATinsight.com I Call us : +91-9999687183 / 9891333772
Online One-on-One Skype based classes and Classroom Coaching in South and West Delhi
http://www.GMATinsight.com/testimonials.html

22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION

Intern
Joined: 07 Jan 2016
Posts: 1
In the xy-plane, point (r, s) lies on a circle with center  [#permalink]

### Show Tags

07 Jan 2016, 12:01
I'm quoting Bunuel's correct explanation below with some follow up questions, since this question's wording did not seem precise to me:

1. The given information says that point (r,s) lies on the circle. For future reference, will "on" mean on the circumference of the circle? In other words, "on" can never refer to a point inside the circle. If the point (r,s) were some arbitrary point inside of the circle, the solution to this problem would be incorrect I believe.

2. Is the point (r,s) assumed to NOT be constant? It seems like the explanation for working out (2) depends upon the fact that (r,s) is non-constant and could be ANY point along the circumference of the circle. If (r,s) were in fact constant, then I don't believe (2) would provide any information in deducing what r^2+s^2 is.

EDIT: I missed the above clarification regarding the language for "on the circle". I have crossed that question of mine out. Sorry!

---------------------------------------------------------------------------------------------------------
Bunuel wrote:
DeeptiM wrote:
OA is D...can anyone explain??

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.

(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.

Hope it helps.
Current Student
Joined: 20 Mar 2014
Posts: 2643
Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44
GPA: 3.7
WE: Engineering (Aerospace and Defense)
Re: In the xy-plane, point (r, s) lies on a circle with center  [#permalink]

### Show Tags

07 Jan 2016, 12:44
lillylw wrote:
I'm quoting Bunuel's correct explanation below with some follow up questions, since this question's wording did not seem precise to me:

1. The given information says that point (r,s) lies on the circle. For future reference, will "on" mean on the circumference of the circle? In other words, "on" can never refer to a point inside the circle. If the point (r,s) were some arbitrary point inside of the circle, the solution to this problem would be incorrect I believe.

2. Is the point (r,s) assumed to NOT be constant? It seems like the explanation for working out (2) depends upon the fact that (r,s) is non-constant and could be ANY point along the circumference of the circle. If (r,s) were in fact constant, then I don't believe (2) would provide any information in deducing what r^2+s^2 is.

EDIT: I missed the above clarification regarding the language for "on the circle". I have crossed that question of mine out. Sorry!

For analysing statement 2, it does not matter whether (r,s) is constant.

Equation of a circle with center (a,b) and radius R is $$(x-a)^2+(y-b)^2=R^2$$ , now as the given circle is centered at (0,0) ---> a=b=0 ---> the equation of the circle thus becomes $$x^2+y^2 = R^2$$

As (r,s) lies on the circle ---> you can substitute r for x and s for y in the equation of the circle to get, $$r^2+s^2=R^2$$

Per statement 2, [$$\sqrt{2} , -\sqrt{2}$$] lies on the circle ---> from equation of the circle $$x^2+y^2 = R^2$$ ---> $$(\sqrt{2})^2+(-\sqrt{2})^2=R^2$$

But as mentioned above, $$r^2+s^2=R^2$$ ---> $$r^2+s^2 =(\sqrt{2})^2+(-\sqrt{2})^2 = 4$$ = a unique value.

Thus, it does not matter what particular values r and s take. Whatever set of (r,s) values we will get, will always satisfy the equation $$r^2+s^2=4$$. Some of the sets can be

$$(\sqrt{2}), -\sqrt{2})$$ or
$$(1, -\sqrt{3})$$ or
$$(1, \sqrt{3})$$ or
$$(-\sqrt{3},1)$$ or
$$(\sqrt{3}, 1)$$ etc

Hope this helps.
Manager
Joined: 29 Dec 2014
Posts: 69
Re: In the xy-plane, point (r, s) lies on a circle with center  [#permalink]

### Show Tags

02 Mar 2017, 21:12
Hi,

A very fundamental or, maybe, a silly question, in GMAT, when a question like this reads 'on a circle', it's supposed to mean, on the circumference of the circle, and not the whole of the area of the circle.

Thanks
Math Expert
Joined: 02 Sep 2009
Posts: 47946
Re: In the xy-plane, point (r, s) lies on a circle with center  [#permalink]

### Show Tags

03 Mar 2017, 01:40
WilDThiNg wrote:
Hi,

A very fundamental or, maybe, a silly question, in GMAT, when a question like this reads 'on a circle', it's supposed to mean, on the circumference of the circle, and not the whole of the area of the circle.

Thanks

Yes, on the circle means on the circumference.
In the circle means within.
_________________
Senior Manager
Joined: 19 Apr 2016
Posts: 274
Location: India
GMAT 1: 570 Q48 V22
GMAT 2: 640 Q49 V28
GPA: 3.5
WE: Web Development (Computer Software)
Re: In the xy-plane, point (r, s) lies on a circle with center  [#permalink]

### Show Tags

03 Mar 2017, 01:51
Bunuel wrote:
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:
$$(x-a)^2+(y-b)^2=r^2$$

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: http://gmatclub.com/forum/math-coordina ... 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.

(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.

Hope it helps.

Another way of using St II is since (r,s) and $$(\sqrt{2}, \ -\sqrt{2})$$ lie on the circle
Their distance from the center(origin in this case) will be same

Therefore, $$r^2+s^2= (\sqrt{2})^2 + ( -\sqrt{2})^2$$ = 2+2 = 4
Manager
Joined: 29 Dec 2014
Posts: 69
Re: In the xy-plane, point (r, s) lies on a circle with center  [#permalink]

### Show Tags

03 Mar 2017, 08:20
Bunuel: thanks for clarifying!
Intern
Joined: 08 Jul 2017
Posts: 4
Re: In the xy-plane, point (r, s) lies on a circle with center  [#permalink]

### Show Tags

20 Jul 2017, 23:09
So, for statement 2, any coordinates that are on the circle can be used to substitute for (r,s)?
Math Expert
Joined: 02 Sep 2009
Posts: 47946
Re: In the xy-plane, point (r, s) lies on a circle with center  [#permalink]

### Show Tags

20 Jul 2017, 23:50
1
parkerd wrote:
So, for statement 2, any coordinates that are on the circle can be used to substitute for (r,s)?

Yes, if a circle is centred at the origin, then the x and y-coordinates of any point on the circle (so on the circumference) will satisfy $$x^2+y^2=radius^2$$
_________________
Intern
Joined: 08 Jul 2017
Posts: 4
Re: In the xy-plane, point (r, s) lies on a circle with center  [#permalink]

### Show Tags

21 Jul 2017, 00:09
If the (r,s) was inside the circle would you still be able to solve with the same equation like in statement 1? Or be able to substitute for (r,s) with the given values on the circle like in statement 2?
Math Expert
Joined: 02 Sep 2009
Posts: 47946
Re: In the xy-plane, point (r, s) lies on a circle with center  [#permalink]

### Show Tags

21 Jul 2017, 00:26
parkerd wrote:
If the (r,s) was inside the circle would you still be able to solve with the same equation like in statement 1? Or be able to substitute for (r,s) with the given values on the circle like in statement 2?

If (r, s) were IN the circle, then the answer would be E because each statement gives us basically the same info - the length of the radius. How can we find the sum of the squares of a random point inside the circle just knowing the radius?
_________________
Re: In the xy-plane, point (r, s) lies on a circle with center &nbs [#permalink] 21 Jul 2017, 00:26

Go to page    1   2    Next  [ 21 posts ]

Display posts from previous: Sort by

# Events & Promotions

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.