# How do you find the center, radius, and equation of a circle with : (1,-2) and (7,6) as the diameter's endpoints?

Jan 26, 2016

${\left(x - 4\right)}^{2} + {\left(y - 2\right)}^{2} = 25$

#### Explanation:

The centre of the circle $c$ is at the midpoint of the diameter, and the radius $r$is equal to half the diameter. We therefore need the distance between the two points.

Using Pythagoras to calculate the length of the diameter $x y$ gives
${\left(x y\right)}^{2} = {\left(7 - 1\right)}^{2} + {\left(6 - \left(- 2\right)\right)}^{2}$
${\left(x y\right)}^{2} = 36 + 64 = 100$

$\therefore y x = \sqrt{100} = 10$

The radius is therefore $5$

$c$ is at the midpoint of $x y$. Its $x$ coordinate $h$ is therefore midway between $x$ and $z$, and its $y$ coordinate $k$ is midway between $z$ and $y$.
$h = 1 + \frac{7 - 1}{2} = 4$
$k = - 2 + \frac{6 - \left(- 2\right)}{2} = 2$

The equation of the circle is therefore ${\left(x - 4\right)}^{2} + {\left(y - 2\right)}^{2} = 25$