# How do you write the equation of the circle with a diameter that has endpoints at (7, –4) and (1, –10)?

Apr 11, 2016

${\left(x - 4\right)}^{2} + {\left(y + 7\right)}^{2} = 18$

#### Explanation:

The standard form of the equation of a circle is.

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{{\left(x - a\right)}^{2} + {\left(y - b\right)}^{2} = {r}^{2}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

where (a ,b ) are the coordinates of the centre and r , the radius.

To obtain the equation , we require to find it's centre and radius.

Given the endpoints of the diameter , the centre will be at the mid-point and the radius will be the distance from the centre to either one of the endpoints.

Given 2 points $A \left({x}_{1} , {y}_{1}\right) \text{ and "B (x_2,y_2) " the mid-point is }$

$\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{{M}_{A B} = \frac{1}{2} \left({x}_{1} + {x}_{2}\right) , \frac{1}{2} \left({y}_{1} + {y}_{2}\right)} |}}}$

Here the 2 endpoints are (7,-4) and (1,-10)

hence centre = $\left[\frac{1}{2} \left(7 + 1\right) , \frac{1}{2} \left(- 4 - 10\right)\right] = \left(4 , - 7\right)$

To calculate the radius: distance from (4,-7) to (7,-4) using the $\textcolor{b l u e}{\text{ distance formula }}$

$\textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
where $\left({x}_{1} , {y}_{1}\right) \text{ and " (x_2,y_2)" are 2 coordinate points }$

let $\left({x}_{1} , {y}_{1}\right) = \left(7 , - 4\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(4 , - 7\right)$

$\Rightarrow \text{ radius } = \sqrt{{\left(4 - 7\right)}^{2} + {\left(- 7 + 4\right)}^{2}} = \sqrt{9 + 9}$
$= \sqrt{18}$

We now have centre = (4 , -7) and radius =$\sqrt{18}$
hence equation of circle:

${\left(x - 4\right)}^{2} + {\left(y + 7\right)}^{2} = 18$