Solution:
Given that,
The diameter of a circle has endpoints (-2,-6) and (8,-10)
Use the midpoint formula to find center of circle
[tex]C = (\frac{x_1 + x_2}{2} , \frac{y_1+y_2}{2})\\\\C = (\frac{ -2+8}{2}, \frac{-6-10}{2})\\\\C = (3, -8)[/tex]
Radius = distance between center and one of end points
distance between (3 , -8) and (-2 , -6)
Use distance formula
[tex]r = \sqrt{(x2-x1)^2 + (y2-y1)^2}[/tex]
[tex]r = \sqrt{(-2-3)^2 + (-6+8)^2}\\\\r = \sqrt{25 + 4}\\\\r = \sqrt{29}[/tex]
The standard equation of the circle:
[tex](x - h)^2 + (y-k)^2 = r^2[/tex]
Where, (h , k) is the center of circle
[tex](x - 3)^2 + (y + 8)^2 = 29[/tex]
Thus the equation of circle is found