Question

Find an equation of the circle whose diameter has endpoints (6,4) and (2,-6)

Answer 1

`(x-4)^2+(y+1)^2=29`

Explanation:

We want to find a circle whose diameter has the endpoints (6, 4) and (2, -6).

Since this is the diameter, its midpoint will be the center of the circle. Find the midpoint:

`\displaystyle M=\left((6+2)/(2), \frac{4+(-6)}2}\right)=(4, -1)`

So, the center of our circle is (4, -1).

Next, to find the radius, we can find the length of the diameter and divide it by half.

Using the distance formula, find the length of the diameter:

`\begin{aligned} d&=√((x_2-x_1)^2+(y_2-y_1)^2)\\\\ &=√((2-6)^2+(-6-4)^2)\\\\&=√((4)^2+(-10)^2)\\\\&=√(116)\\\\&=2√(29)\end{aligned}`

So, the radius will be:

`\displaystyle r=(1)/(2)d=(1)/(2)\left(2√(29)\right)=√(29)`

The equation for a circle is given by:

`\displaystyle (x-h)^2+(y-k)^2=r^2`

Substitute:

`(x-4)^2+(y+1)^2=29`