Question

Find the equation of tangent to the circle x² + y² − 4x + 3y + 2 = 0 at the point (4, −2)

Answer 1

The equation of tangent to the circle

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

xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0  ...(1)

Comparing the equation x² + y² – 4x + 3y + 2 = 0 with

x² + y² + 2gx + 2fy + c = 0, we get,

2g = – 4, 2f = 3 and c = 2

∴ g = `– 2, "f" = 3/2 and "c" = 2`

∴ from (1), the equation of tangent to the given circle at (4, –2) is

`x(4) + y(-2) -2(x + 4) + 3/2(y - 2) + 2` = 0

∴ `4x - 2y - 2x - 8 + 3/2y - 3 + 2` = 0

∴ `2x - 1/2y - 9` = 0

∴ 4x – y – 18 = 0.