Question

Show that x = −1 is a tangent to circle x² + y² − 2y = 0 at (−1, 1).

Answer 1

We have to find the equation of tangent to the circle x² + y² – 2y = 0 at (– 1, 1).

The equation of the tangent to the circle

x² + y² + 2gx + 2fy + c = 0 at (x₁, y₁) is

xx₁ +yy₁ + g(x + x₁) + f(y + y₁) + c = 0

Comparing the equation x² + y² – 2y = 0 with

x² + y² + 2gx + 2fy + c = 0, we get,

g = 0, f = – 1, c = 0

∴ The equation of the tangent to the given circle at (– 1, 1) is

x(–1) + y(1) – 0(x – 1) – 1(y + 1) + 0 = 0

∴ –x + y – y – 1 = 0

∴ x = – 1

Hence, x = – 1 is tangent to the circle x² + y² – 2y = 0 at (– 1, 1).