Question

find an equation of the circle that has center (-4, 3) and passes through (-2, -5).

Explanation:

Step1: Calculate the radius

The radius $r$ is the distance between the center $(-4,3)$ and the point $(-2,-5)$. Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, $x_1=-4,y_1 = 3,x_2=-2,y_2=-5$. So $r=\sqrt{(-2+4)^2+(-5 - 3)^2}=\sqrt{2^2+(-8)^2}=\sqrt{4 + 64}=\sqrt{68}$.

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. Here, $h=-4,k = 3,r=\sqrt{68}$. So the equation is $(x + 4)^2+(y - 3)^2=68$.

Answer:

$(x + 4)^2+(y - 3)^2=68$