Question

In the diagram, a circle centered at the origin, a right triangle, and the Pythagorean theorem are used to derive the equation of a circle, x2 + y2 = r2.

On a coordinate plane, circle Q is centered at the origin with radius r. Triangle P Q S is shown. Point Q is at (0, 0) and points P is at (x, y). The length of Q P is r, the length of P S is y, and the length of Q S is x. Angle P S Q is a right angle.

If the center of the circle were moved from the origin to the point (h, k) and point P at (x, y) remains on the edge of the circle, which could represent the equation of the new circle?

(h + x)2 + (k + y)2 = r2
(x – h)2 + (y – k)2 = r2
(k + x)2 + (h + y)2 = r2
(x – k)2 + (y – h)2 = r2

Answer 1

(x – h)² + (y – k)² = r²

Reason: the distance from point (x,y) to center (h,k) is sqrt((x−h)²+(y−k)²), which equals r for any point on the circle, so (x−h)²+(y−k)²=r².