What is the equation of the circle with endpoints of the diameter of a circle are (1,-1) and (9,5)?

1 Answer
Jan 4, 2016

${\left(x - 5\right)}^{2} + {\left(y - 2\right)}^{2} = 25$

Explanation:

A general circle centred at $\left(a , b\right)$ and having radius $r$ has equation ${\left(x - a\right)}^{2} + {\left(y - b\right)}^{2} = {r}^{2}$.

The centre of the circle would be the midpoint between the 2 diameter endpoints, ie $\left(\frac{1 + 9}{2} , \frac{- 1 + 5}{2}\right) = \left(5 , 2\right)$

The radius of the circle would be half the diameter, ie. half the distance between the 2 points given, that is

$r = \frac{1}{2} \left(\sqrt{{\left(9 - 1\right)}^{2} + {\left(5 + 1\right)}^{2}}\right) = 5$

Thus the equation of the circle is

${\left(x - 5\right)}^{2} + {\left(y - 2\right)}^{2} = 25$.