$\text{The end points of a diameter in a circle is given by the points }P\left(-1,0\right)\text{ and }Q\left(1,1\right).$

$\text{Find the equation of the circle.}$

Answer

$\text{The center of the circle is the midpoint of any of its diameter}$

$\text{So, the center of the circle is given by the point }O\left(0,\frac{1}{2}\right)$

$\text{Using the distance formula on the end points of the diameter,}$

$\text{We find the diameter }=\sqrt{5}$

$\text{The radius of the circle is half the diameter}$

$\text{So, the radius of the circle }=\sqrt{\frac{5}{4}}$$\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:}$

$\left(x-x_0\right)^2 + \left(y-y_0\right)^2=k^2$.

$\text{or, }\left(x \right)^2 + \left(y -\frac{1}{2}\right)^2=\left(\sqrt{\frac{5}{4}}\right)^2$

$\text{or, }x^2+y^2-y+\frac{1}{4}=\frac{5}{4}$

$\text{or, } x^2 + y^2-1y-1=0$