Question

Given A = (-1,0) and B = (11,4), how do you show that the equation of the circle with AB as diameter may be written as `(x-5)^2 + (y-2)^2 = 40`?

Answer 1

see explanation.

Explanation:

Recall that the center-radius form of the circle equation is in the format:
`(x – h)^2 + (y – k)^2 = r^2`,
with the center being at the point `(h, k)` and the radius being `r`.

Given `A(-1,0), and B(11,4)`,
Let distance between `A and B` be `D`.
`=> D=sqrt((11-(-1))^2+(4-0)^2)=sqrt160=sqrt(16*10)=4sqrt10`
Given `D` is the diameter, `=> D/2 = R` (radius)
`=> R=D/2=(4sqrt10)/2=2sqrt10`
The circle is centered at midpoint of AB :
`=>` midpoint of `AB= ((-1+11)/2, (0+4)/2)=(5,2)`

So the equation of the circle can be written as :
`(x-5)^2+(y-2)^2=(2sqrt10)^2`,
`=> (x-5)^2+(y-2)^2=40` (proved)