Points (2, 1) and (6, 7) are opposite…

Question 1 Report

Points (2, 1) and (6, 7) are opposite vertices of a square which is inscribed in a circle. Find the :

(a) centre of the circle ; (b) equation of the circle.

Answer

Answer Details

Square inscribed in a circle with opposite vertices \((2,1)\) and \((6,7)\).

Opposite vertices of the square are the ends of a diagonal, which is also a diameter of the circle.

(a) Centre = midpoint of the diagonal:

\[\left(\frac{2+6}{2},\frac{1+7}{2}\right)=(4,4)\]

(b) Equation. The diameter length is

\[d=\sqrt{(6-2)^{2}+(7-1)^{2}}=\sqrt{16+36}=\sqrt{52}\]

so the radius is \(r=\dfrac{\sqrt{52}}{2}=\sqrt{13}\) and \(r^{2}=13\). The equation of the circle is

\[(x-4)^{2}+(y-4)^{2}=13\]

Points (2, 1) and (6, 7) are opposite vertices of a square which is inscribed in a circle. Find the : (a) centre of the circle ; (b) equation of the circle.