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, ( x^{2}+y^{2}=r^{2} ). 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}=r^{2} ) ( (x - h)^{2}+(y - k)^{2}=r^{2} ) ( (k + x)^{2}+(h + y)^{2}=r^{2} ) ( (x - k)^{2}+(y - h)^{2}=r^{2} )

Explanation:

Step1: Recall the distance formula

The distance \(d\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

Step2: Apply the distance formula for the circle

The radius \(r\) of a circle is the distance between the center \((h,k)\) and a point \((x,y)\) on the circle. So \(r=\sqrt{(x - h)^2+(y - k)^2}\).

Step3: Square both sides of the equation

Squaring both sides gives \((x - h)^2+(y - k)^2=r^2\).

Answer:

\((x - h)^2+(y - k)^2 = r^2\) (the second option)