Question

find the center and radius of the circle. write the standard form of the equation.

Explanation:

Step1: Find the center of the circle

The center of a circle is the mid - point of a diameter. Given two points on a diameter $(x_1,y_1)=(3,8)$ and $(x_2,y_2)=(12,8)$. The mid - point formula is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. So, the x - coordinate of the center is $\frac{3+12}{2}=\frac{15}{2}=7.5$, and the y - coordinate is $\frac{8 + 8}{2}=8$. The center is $(7.5,8)$.

Step2: Find the radius of the circle

The radius is the distance from the center to a point on the circle. Using the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$ with the center $(7.5,8)$ and the point $(12,8)$. Here, $x_1 = 7.5,y_1 = 8,x_2=12,y_2 = 8$. Then $r=\sqrt{(12 - 7.5)^2+(8 - 8)^2}=\sqrt{(4.5)^2+0^2}=4.5$.

Step3: Write the standard form of the equation

The standard form of the equation of a circle is $(x - h)^2+(y - k)^2=r^2$, where $(h,k)$ is the center and $r$ is the radius. Substituting $h = 7.5,k = 8,r = 4.5$ we get $(x - 7.5)^2+(y - 8)^2=(4.5)^2=20.25$.

Answer:

Center: $(7.5,8)$; Radius: $4.5$; Standard form: $(x - 7.5)^2+(y - 8)^2=20.25$