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

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Joined: 06 Apr 2010
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In the xy-plane, point (r, s) lies on a circle with center [#permalink]  09 Dec 2010, 09:31
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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.
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Kudos [?]: 40967 [1] , given: 5576

Re: Point on a circle [#permalink]  09 Dec 2010, 09:45
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udaymathapati 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.

THEORY:
In an x-y Cartesian coordinate system, 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:
$$x^2+y^2=r^2$$

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.

(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.

_________________
Re: Point on a circle   [#permalink] 09 Dec 2010, 09:45
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