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Question$\text{The end points of a diameter in a circle is given by the points }P\left(-2,-8\right)\text{ and }Q\left(-8,6\right).$
$\text{Find the equation of the circle.}$
Answer$\text{The center of the circle is the midpoint of any of its diameter}$
$\text{So, the center of the circle is given by the point }O\left(-5,-1\right)$
$\text{Using the distance formula on the end points of the diameter,}$
$\text{We find the diameter }=\sqrt{232}$
$\text{The radius of the circle is half the diameter}$
$\text{So, the radius of the circle }=\sqrt{58}$$\text{The equation of the circle with center at }O\left(x_0,y_0\right)\text{ and radius }r=k \text{ is given by:}$
$\left(x-x_0\right)^2 + \left(y-y_0\right)^2=k^2$.
$\text{or, }\left(x +5\right)^2 + \left(y +1\right)^2=\left(\sqrt{58}\right)^2$
$\text{or, }x^2+10x+25+y^2+2y+1=58$
$\text{or, } x^2 + y^2+10x+2y-32=0$