Question

If y = 2x is a chord of circle x² + y²−10x = 0, find the equation of circle with this chord as diametre

Answer 1

The equation of the circle is

x² + y² − 10x = 0   ...(1)

and the equation of the line is y = 2x  ...(2)

Let A(x₁, y₁) and B(x₂, y₂) be the points of intersection of circle and the line.

To find the points of intersection, substitute y = 2x in 1 equation (1), we get,

x² + 4x² − 10x = 0

∴ 5x² − 10x = 0

∴ x(5x − 10) = 0

∴ x = 0 or x = 2

∴ x₁ = 0 and x₂ = 2

Also, y₁ = 2x₁ and y₂ = 2x₂

∴ y₁ = 0 and y₂ = 4

∴ A ≡ (0, 0) and B ≡ (2, 4)

∴ by diameter form, the equation of the circle on chord AB as diameter is

(x − 0)(x − 2) + (y − 0)(y − 4) = 0

∴ x² − 2x + y² − 4y = 0

∴ x² + y² − 2x − 4y = 0.