Question

find the equation for the circle with center (-3,4) and passing through (-1,4). write the equation for the circle. (simplify your answer.)

Explanation:

Step1: Recall circle - equation formula

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 and $r$ is the radius. Here, the center is $(-3,4)$, so $h=-3$ and $k = 4$.

Step2: Calculate the radius

The radius $r$ is the distance between the center $(-3,4)$ and the point $(-1,4)$ on the circle. Using the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$, we have $x_1=-3,y_1 = 4,x_2=-1,y_2 = 4$. Then $r=\sqrt{(-1-(-3))^2+(4 - 4)^2}=\sqrt{(-1 + 3)^2+0^2}=\sqrt{2^2}=2$.

Step3: Write the equation of the circle

Substitute $h=-3,k = 4,r = 2$ into the standard - form equation $(x - h)^2+(y - k)^2=r^2$. We get $(x+3)^2+(y - 4)^2=4$.

Answer:

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