Question

If the lines x + y = 6 and x + 2y = 4 are diameters of the circle, and the circle passes through the point (2, 6) then find its equation.

Answer 1

To get coordinates of centre we should solve the equations of the diameters x + y = 6, x + 2y = 4.

x + y = 6 ……. (1)

x + 2y = 4 ………. (2)

(1) – (2) ⇒ -y = 2

y = -2

Using y = -2 in (1) we get x – 2 = 6

x = 8

Centre is (8, -2) the circle passes through the point (2, 6).

∴ Radius = `sqrt((8 - 2)^2 + (- 2 - 6)^2)`

`= sqrt(6^2 + (- 8)^2)`

`= sqrt(36 + 64)`

`= sqrt100` = 10

Equation of the circle with centre (h, k) and radius r is (x – h)² + (y – k)² = r²

⇒ (x – 8)² + (y + 2)² = 10²

⇒ x² + y² – 16x + 4y + 64 + 4 = 100

⇒ x² + y² – 16x + 4y – 32 = 0