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

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In the xy-plane, point (r, s) lies on a circle with center a [#permalink]  07 Jan 2014, 04:17
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The Official Guide For GMAT® Quantitative Review, 2ND Edition

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

Data Sufficiency
Question: 22
Category: Geometry Simple coordinate geometry
Page: 154
Difficulty: 600

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Math Expert
Joined: 02 Sep 2009
Posts: 27457
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Kudos [?]: 42071 [0], given: 5945

Re: In the xy-plane, point (r, s) lies on a circle with center a [#permalink]  07 Jan 2014, 04:17
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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: 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 $$x^2+y^2=r^2$$ 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.

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Re: In the xy-plane, point (r, s) lies on a circle with center a [#permalink]  07 Jan 2014, 21:33
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 a [#permalink]  08 Jan 2014, 08:04
Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

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

The formula of circle is given by (x-a)^2 + (y-b)^2 = r^2 where circle is centered as (a,b) and r = radius of the circle.

So if the circle is centered at origin, then the equation reduces to x^2 + y^2 = r^2.

Statement 1) (r,s) lies on the circle and hence r^2 + s^2 should be equal to the radius of the circle and the statement provides that very fact. Hence Sufficient.
Statement 2) $$(\sqrt{2}, \ -\sqrt{2})$$ lies on the circle. Then the radius can be calculated as root(root(2)^2) + root(2)^2) = 2 and as the radius is same from any point on the circle. Hence Sufficient.

Option D)
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Re: In the xy-plane, point (r, s) lies on a circle with center a [#permalink]  11 Jan 2014, 06:14
Expert's post
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: 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 $$x^2+y^2=r^2$$ 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.

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Re: In the xy-plane, point (r, s) lies on a circle with center a [#permalink]  27 Jan 2015, 23:00
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Re: In the xy-plane, point (r, s) lies on a circle with center a   [#permalink] 27 Jan 2015, 23:00
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# In the xy-plane, point (r, s) lies on a circle with center a

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