Question

fill in each blank so that the resulting statement is true. the standard form of the equation of a circle with center $(h,k)$ and radius $r$ is
the standard form of the equation of a circle with center $(h,k)$ and radius $r$ is

$$left( \frac { x } { h } ight) ^ { 2 } + left( \frac { y } { k } ight) ^ { 2 } = r ^ { 2 }$$

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

Explanation:

Step1: Recall the distance formula

The distance \(d\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For a circle, the distance between the center \((h,k)\) and any point \((x,y)\) on the circle is the radius \(r\).

Step2: Apply the distance formula

Set \(d = r\), \(x_1=h\), \(y_1 = k\), \(x_2=x\), \(y_2=y\). Then \(r=\sqrt{(x - h)^2+(y - k)^2}\). Squaring both sides gives \((x - h)^2+(y - k)^2=r^2\).

Answer:

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