Question

a circle with its center at (m) passes through point (p) and is graphed in the (xy - plane). which of the following is an equation for the graph of the circle?

Explanation:

Step1: Recall the standard - form of a circle equation

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, the center of the circle $M$ has coordinates $(h,k)=(0,1)$.

Step2: Calculate the radius

The radius $r$ is the distance between the center $M(0,1)$ and the point $P(3,5)$ on the circle. Use the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Substitute $x_1 = 0,y_1 = 1,x_2 = 3,y_2 = 5$ into the formula: $r=\sqrt{(3 - 0)^2+(5 - 1)^2}=\sqrt{3^2+4^2}=\sqrt{9 + 16}=\sqrt{25}=5$.

Step3: Write the circle equation

Substitute $h = 0,k = 1,r = 5$ into the standard - form of the circle equation $(x - h)^2+(y - k)^2=r^2$. We get $(x-0)^2+(y - 1)^2=5^2$, which simplifies to $x^{2}+(y - 1)^{2}=25$.

Answer:

$x^{2}+(y - 1)^{2}=25$