The equation of the circle is [tex](x-2)^{2} +(y-1)^{2} = 13[/tex].
What is a circle?
A circle is a collection of all points in a plane which are at a constant distance from a fixed point. A circle is a round-shaped figure that has no corners or edges.
For the given situation,
The endpoints of the diameter of a circle are P = (x1,y1) is (-1,-1) and Q = (x2,y2) is (5,3).
The general form of equation of the circle is
[tex](x-h)^{2} +(y-k)^{2} =r^{2}[/tex],
where (h,k) is the center of the circle and r is the radius of the circle.
Diameter of the circle can be found using the distance formula,
[tex]d=\sqrt{(x2-x1)^{2}+(y2-y1)^{2} }[/tex]
⇒ [tex]d=\sqrt{(5-(-1))^{2}+(3-(-1))^{2} }[/tex]
⇒ [tex]d=\sqrt{(5+1)^{2}+(3+1)^{2} }[/tex]
⇒ [tex]d=\sqrt{(6)^{2}+(4)^{2} }[/tex]
⇒ [tex]d=\sqrt{36+16 }[/tex]
⇒ [tex]d=\sqrt{52}[/tex]
⇒ [tex]d=7.2111[/tex] ≈ [tex]7.2[/tex]
Radius,r = [tex]\frac{diameter}{2}[/tex]
⇒ [tex]r=\frac{7.2}{2}[/tex]
⇒ [tex]r=3.6[/tex]
The center of the circle can be found by using the mid point formula,
[tex](h,k)=(\frac{x1+x2}{2} ,\frac{y1+y2}{2} )[/tex]
⇒ [tex](h,k)=(\frac{-1+5}{2} ,\frac{-1+3}{2} )[/tex]
⇒ [tex](h,k)=(\frac{4}{2} ,\frac{2}{2} )[/tex]
⇒ [tex](h,k)=(2,1)[/tex]
Thus the equation of circle becomes,
⇒ [tex](x-2)^{2} +(y-1)^{2} =3.6^{2}[/tex]
⇒ [tex](x-2)^{2} +(y-1)^{2} = 12.97[/tex]
⇒ [tex](x-2)^{2} +(y-1)^{2} = 13[/tex]
Hence we can conclude that the equation of the circle is [tex](x-2)^{2} +(y-1)^{2} = 13[/tex].
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