Question

A circle has a diameter with endpoints (7, -7) and (5, -3). What is the equation of the circle?

Answer 1

For equation, we need 1) center, 2) radius
both can be found by using distance formula between 2 points:
$d = \sqrt{ {(x2 - x1)}^{2} + {(y2 - y1)}^{2} } \\ = \sqrt{ {(5 - 7)}^{2} + {( - 3 - - 7)}^{2} } \\ = \sqrt{ {(2)}^{2} + {(4)}^{2} } = \sqrt{4 + 16}$
$d = \sqrt{20} = \sqrt{4} \times \sqrt{5} = 2 \sqrt{5}$
So our diameter is 2sr5, therefore our radius is half that: r = d/2 = 2sr5/2 = sr5
Now to calculate the center, go halfway between the x's and between the y's:
((7-5)/2 + 5, (-7--3)/2 + -3)
= ((2)/2 + 5, (-7+3)/2 + -3) = (1 + 5, (-4)/2 + -3)
= (1 + 5, -2 + -3) = (6, -5)
So our center is at (6, -5), where (h, k) is center for the formula, so h = 6, k = -5
Equation of a circle:
${(x - h)}^{2} + {(y - k)}^{2} = {r}^{2} \\ {(x - 6)}^{2} + {(y + 5)}^{2} = {( \sqrt{5}) }^{2} \\ {(x - 6)}^{2} + {(y + 5)}^{2} = 5$