# How do you write an equation for a circle with Point A is (4, -2) Point B is (10,6) as diameters?

May 30, 2016

We must first find the length of the diameter and the center of the circle.

The center is an equal distance from all points inside a circle. Therefore, we can use the midpoint formula $\left(\frac{{x}_{1} + {x}_{2}}{2} , \frac{{y}_{1} + {y}_{2}}{2}\right)$ to find the center.

$\left(\frac{{x}_{1} + {x}_{2}}{2} , \frac{{y}_{1} + {y}_{2}}{2}\right)$

$= \left(\frac{4 + 10}{2} , \frac{- 2 + 6}{2}\right)$

$= \left(7 , 2\right)$

The center will therefore be at $\left(7 , 2\right)$.

Now for the length of the diameter.

This can be found by using the distance theorem, a simple variation on pythagorean theorem.

$d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

$d = \sqrt{{6}^{2} + {8}^{2}}$

$d = \sqrt{100}$

$d = 10$

Hence, the diameter measures $10$ units. Since the equation of the circle is of the form ${\left(x - h\right)}^{2} + {\left(y - k\right)}^{2} = {r}^{2}$, where $\left(h , k\right)$ is the center and $r$ is the radius, we need the radius, and not the diameter. The equation $d = 2 r$ shows the relationship between the diameter (d) and the radius (r).

Solving for r:

$r = \frac{d}{2}$

$r = \frac{10}{2}$

$r = 5$

Now that we know our radius, we can substitute what we know into the equation of the circle, of the form mentioned above:

${\left(x - 7\right)}^{2} + {\left(y - 2\right)}^{2} = 25$

Here is the graph of this relation (note: it's not a function, since every value of x is not only with one value of y)

Practice exercises:

1. Determine the equation of the circle who's diameter ends at the points $\left(- 1 , - 4\right)$ and $\left(3 , - 5\right)$.

$2.$ Determine the equation of the following circle.

Hopefully this helps, and good luck!