Question

find the equation for the circle with center (-4,4) and passing through (3, - 4). write the equation for the circle. (simplify your answer.)

Explanation:

Step1: Recall the circle - equation formula

The standard - form equation of a circle 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 $(h,k)=(-4,4)$.

Step2: Calculate the radius

The radius $r$ is the distance between the center $(-4,4)$ and the point $(3,-4)$ on the circle. Use the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
Let $(x_1,y_1)=(-4,4)$ and $(x_2,y_2)=(3,-4)$. Then $r=\sqrt{(3+4)^2+(-4 - 4)^2}=\sqrt{7^2+(-8)^2}=\sqrt{49 + 64}=\sqrt{113}$.

Step3: Write the equation of the circle

Substitute $h=-4$, $k = 4$, and $r=\sqrt{113}$ into the standard - form equation of the circle:
$(x+4)^2+(y - 4)^2=113$.

Answer:

$(x + 4)^2+(y - 4)^2=113$