Question

the endpoints of a diameter of a circle are (-6, 4) and (-22, -8). (a) write an equation of the circle in standard form. (b) graph the circle. part: 0 / 2 part 1 of 2 (a) an equation of the circle in standard form is

Explanation:

Step1: Find the center of the circle

The center of the circle is the mid - point of the diameter. The mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Here, $x_1=-6,y_1 = 4,x_2=-22,y_2=-8$. So the center $(h,k)=(\frac{-6+( - 22)}{2},\frac{4+( - 8)}{2})=(\frac{-28}{2},\frac{-4}{2})=(-14,-2)$.

Step2: Find the radius of the circle

The radius $r$ is the distance from the center $(h,k)$ to either of the endpoints of the diameter. Using the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$, with $(h,k)=(-14,-2)$ and $(x_1,y_1)=(-6,4)$. Then $r=\sqrt{(-6+14)^2+(4 + 2)^2}=\sqrt{8^2+6^2}=\sqrt{64 + 36}=\sqrt{100}=10$.

Step3: Write the standard - form equation of the circle

The standard - form equation of a circle is $(x - h)^2+(y - k)^2=r^2$. Substituting $h=-14,k=-2,r = 10$ into the equation, we get $(x + 14)^2+(y + 2)^2=100$.

Answer:

$(x + 14)^2+(y + 2)^2=100$

For part (b), to graph the circle:

1. Plot the center of the circle at the point $(-14,-2)$ on the coordinate plane.
2. From the center, move 10 units in all directions (up, down, left, right) to mark points on the circle. Then sketch the circle passing through these points.