Question

One diameter of the circle circumscribing the rectangle ABCD is 4y = x + 7. If the coordinates of A and B are (−3, 4) and (5, 4) respectively, find the equation of the circle.

Answer 1

Clearly, the centre of the circle lies on the line 4y = x + 7.

The circle passes through A (−3, 4) and B (5, 4).

The slope of the segment joining A and B is zero.

Therefore, the slope of the perpendicular bisector of AB is not defined.

Hence, the perpendicular bisector of AB will be parallel to the y-axis and will pass through $\left(\frac{-3+5}{2},\frac{4+4}{2}\right)=\left(1,4\right)$.

The equation of the perpendicular bisector is $x=1$.

The intersection point of the perpendicular bisector and 4y = x + 7 is $\left(1,2\right)$.

∴ Centre = $\left(1,2\right)$

Radius = $\sqrt{{\left(5-1\right)}^2+{\left(4-2\right)}^2}=\sqrt{20}$

Hence, the required equation of the circle is $x^2+y^2-2x-4y-15=0$.