Question

Find the equations of tangents to the parabola y² = 12x from the point (2, 5).

Answer 1

Equation of the parabola is y² = 12x

∴ 4a = 12

∴ a = 3

The equation of the tangent to the parabola with slope m is

`y = mx + a/m`

∴ `y = mx  + 3/m`

If this tangent passes through the point (2, 5), then

5 = `2m + 3/m`

∴ 5m = 2m² + 3

∴ 2m² − 5m + 3 = 0

∴ 2m² − 2m − 3m + 3 = 0

∴ 2m(m − 1) − 3(m − 1) = 0

∴ (m − 1)(2m − 3) = 0

∴ m = 1 or m = 3/2

∴ m₁ = 1 and m₂ = 3/2 are the slopes of the required tangents

∴ the equations of the tangents are

y − 5 = 1(x − 2) and y − 5 = 3/2 (x − 2)

∴ y − 5 = x − 2 and 2y − 10 = 3x − 6

∴ x − y + 3 = 0 and 3x − 2y + 4 = 0