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.
Explanation
(a)
Let O(x, y) be the centre of the circle. O is the midpoint of the two points with given coordinates.
(x = frac{2 + 6}{2} = frac{8}{2} = 4)
(y = frac{1 + 7}{2} = frac{8}{2} = 4)
The centre is (4, 4).
(b) Radius of circle = (sqrt{(4 – 1)^{2} + (4 – 2)^{2}})
= (sqrt{3^{2} + 2^{2}})
= (sqrt{13} units).
Equation of the circle : ((x – 4)^{2} + (y – 4)^{2} = (sqrt{13})^{2})
((x^{2} – 8x + 16) + (y^{2} – 8y + 16) = 13)
(x^{2} + y^{2} – 8x – 8y + 32 – 13 = 0)
(x^{2} + y^{2} – 8x – 8y + 19 = 0)