Question

1. find an equation of the circle that has center c(7, -4) and passes through p(-2, 5).

Explanation:

Step1: Calculate the radius

The radius $r$ is the distance between the center $C(7, - 4)$ and the point $P(-2,5)$. Use the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, $(x_1,y_1)=(7,-4)$ and $(x_2,y_2)=(-2,5)$. So $r=\sqrt{(-2 - 7)^2+(5+4)^2}=\sqrt{(-9)^2+9^2}=\sqrt{81 + 81}=\sqrt{162}=9\sqrt{2}$.

Step2: Write the equation of the circle

The standard - form of the equation of a circle is $(x - h)^2+(y - k)^2=r^2$, where $(h,k)$ is the center of the circle. Given $(h,k)=(7,-4)$ and $r = 9\sqrt{2}$, the equation is $(x - 7)^2+(y + 4)^2=(9\sqrt{2})^2$.

Answer:

$(x - 7)^2+(y + 4)^2 = 162$