Question

the center of a circle and a point on the circle are given. write the equation of the circle in standard form. center (3,2), point on the circle (5, - 5)
the equation of the circle in standard form is
(type an equation. simplify your answer.)

Explanation:

Step1: Recall circle standard - form

The standard form of a circle equation is $(x - h)^2+(y - k)^2=r^2$, where $(h,k)$ is the center of the circle and $r$ is the radius. Here, $h = 3$ and $k = 2$.

Step2: Calculate the radius

The radius $r$ is the distance between the center $(3,2)$ and the point $(5,-5)$ on the circle. Use the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. So, $r=\sqrt{(5 - 3)^2+(-5 - 2)^2}=\sqrt{2^2+( - 7)^2}=\sqrt{4 + 49}=\sqrt{53}$.

Step3: Write the circle equation

Substitute $h = 3$, $k = 2$, and $r=\sqrt{53}$ into the standard - form equation: $(x - 3)^2+(y - 2)^2=53$.

Answer:

$(x - 3)^2+(y - 2)^2=53$