Question

The equation of circle having a diameter with endpoints (-3, 1) and (5, 7) is

Answer 1

Answer: B.) (x - 1)² + (y - 4)² = 25

Answer 2

The equation of a circle:

`(x-h)^2+(y-k)^2=r^2`
(h,k) - the coordinates of the center
r - the radius

The center is the midpoint of the diameter.

`(-3,1) \\ x_1=-3 \\ y_1=1 \\ \\ (5,7) \\ x_2=5 \\ y_2=7 \\ \\ \hbox{the midpoint: } ((x_1+x_2)/(2),(y_1+y_2)/(2)) \\ (x_1+x_2)/(2)=(-3+5)/(2)=(2)/(2)=1 \\ (y_1_y_2)/(2)=(1+7)/(2)=(8)/(2)=4 \\ \Downarrow \\ (1,4) \\ h=1 \\ k=4`

The radius is half the diameter, or the distance from one of the endpoints to the center.

`r=√((x_1-h)^2+(y_1-k)^2)=√((-3-1)^2+(1-4)^2)=\\ =√((-4)^2+(-3)^2)=√(16+9)=√(25)=5 \\ \Downarrow \\ r^2=5^2=25`

The equation of the circle:

`\boxed{(x-1)^2+(y-4)^2=25}`