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

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saurya_s
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In the xy-plane, point (r, s) lies on a circle with center at the orig [#permalink] New post   Updated on: 25 Nov 2020, 17:07
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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


DS02741
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D

Originally posted by saurya_s on 12 Apr 2005, 14:56.
Last edited by bb on 25 Nov 2020, 17:07, edited 6 times in total.
Edited the question and added the OA
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Re: In the xy-plane, point (r, s) lies on a circle with center at the orig [#permalink] New post   11 Feb 2012, 06:30
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SOLUTION

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: https://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 \(x^2+y^2=r^2\) 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 \(x^2+y^2=r^2\) --> \(2+2=4=r^2\). Sufficient.

Answer: D.
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Re: In the xy-plane, point (r, s) lies on a circle with center at the orig [#permalink] New post   03 Mar 2013, 00:02
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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 at the orig [#permalink] New post   02 Mar 2013, 22:08
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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 at the orig [#permalink] New post   07 Jan 2014, 22:33
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If a circle, lying on a xy-plane, has its center at the origin, the equation is x^2+y^2=R^2, where x & y are points on the circle and R is the radius of the circle.

Since x & y from the equation x^2+y^2=R^2 is similar to r & s in the question, we can rewrite as r^2+s^2=R^2.

Statement (1) gives us the value of radius, R; Therefore, r^2+s^2=4; Sufficient.
Statement (2) gives us the value of a point on the circle
=> sub x & y values of the point in r^2+s^2=R^22: 2+2 = 4;
r^2 = 4; Sufficient.

Hence, Answer is (D).
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Re: In the xy-plane, point (r, s) lies on a circle with center at the orig [#permalink] New post   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?
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Re: In the xy-plane, point (r, s) lies on a circle with center at the orig [#permalink] New post   09 Jul 2015, 03:43
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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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Re: In the xy-plane, point (r, s) lies on a circle with center at the orig [#permalink] New post   09 Jul 2015, 03:52
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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 at the orig [#permalink] New post   02 Mar 2017, 21:12
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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
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Re: In the xy-plane, point (r, s) lies on a circle with center at the orig [#permalink] New post   03 Mar 2017, 01:40
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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.
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Re: In the xy-plane, point (r, s) lies on a circle with center at the orig [#permalink] New post   20 Jul 2017, 23:09
So, for statement 2, any coordinates that are on the circle can be used to substitute for (r,s)?
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Re: In the xy-plane, point (r, s) lies on a circle with center at the orig [#permalink] New post   20 Jul 2017, 23:50
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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 \(x^2+y^2=radius^2\)
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Re: In the xy-plane, point (r, s) lies on a circle with center at the orig [#permalink] New post   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?
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Re: In the xy-plane, point (r, s) lies on a circle with center at the orig [#permalink] New post   21 Jul 2017, 00:26
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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?
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Re: In the xy-plane, point (r, s) lies on a circle with center at the orig [#permalink] New post   29 Mar 2020, 04:15
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 \((\sqrt{2}, \ -\sqrt{2})\) lies on the circle


If a point (a, b) is on a circle with center at origin and radius = r, then a^2 + b^2 = r^2

Simply Q: r = ?

1) Sufficient
2) 2 + 2 = 4 = r^2 => Sufficient

ANSWER: D
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Re: In the xy-plane, point (r, s) lies on a circle with center at the orig [#permalink] New post   07 Jul 2020, 20:54
Statement 1 - r and s are two different points on a circle so both have values of 2^2 and 2^ so r^2+s^2 = 8
Is this the correct inference?
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Re: In the xy-plane, point (r, s) lies on a circle with center at the orig [#permalink] New post   08 Nov 2020, 13:09
gayatri259 wrote:
Statement 1 - r and s are two different points on a circle so both have values of 2^2 and 2^ so r^2+s^2 = 8
Is this the correct inference?


That's not correct.

The question stem tells us the point lies on the circle with center at the origin (0,0). If the radius is 2, this means the coordinates could be (2,0), (0,2), etc.)

\(r^2 + s^2\) will always equal 4.
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Re: In the xy-plane, point (r, s) lies on a circle with center at the orig [#permalink] New post   04 Jul 2021, 04:49
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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 \((\sqrt{2}, \ -\sqrt{2})\) lies on the circle


DS02741

Solution:

Question Stem Analysis:

We need to determine the value of r^2 + s^2, given that (r, s) is a point on the circle with center at the origin. Notice that the equation of such a circle is x^2 + y^2 = R^2 where R is the radius of the circle. Since (r, s) is a point on the circle, we have r^2 + s^2 = R^2. That is, in order to determine the value of r^2 + s^2, we either need to know the coordinates of a point on the circle or just the value of R.

Statement One Alone:

Since the radius of the circle is 2, R = 2. Therefore, r^2 + s^2 = 2^2 = 4. Statement one alone is sufficient.

Statement Two Alone:

Since (√2, -√2) is a point on the circle, x = √2 and y = -√2 will satisfy x^2 + y^2 = R^2. Substituting, we find R^2 = x^2 + y^2 = (√2)^2 + (-√2)^2 = 2+ 2 = 4. Proceeding as above, we obtain r^2 + s^2 = 4. Statement two alone is sufficient.

Answer: D
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Re: In the xy-plane, point (r, s) lies on a circle with center at the orig [#permalink] New post   10 Aug 2023, 22:12
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