Question

Answer the following:

The slopes of the tangents drawn from P to the parabola y² = 4ax are m₁ and m₂, show that `("m"_1 /"m"_2)` = k, where k is a constant.

Answer 1

Let P(x₁, y₁) be any point on the parabola y² = 4ax

Equation of tangent to the parabola y² = 4ax having slope m is y = `"m"x + "a"/"m"`

This tangent passes through P(x₁, y₁)

∴ y₁ = `"m"x_1 + "a"/"m"`

∴ my₁ = m²x₁ + a

∴ m²x₁ – my₁ + a = 0

This is a quadratic equation in ‘m’.

The roots m₁ and m₂ of this quadratic equation are the slopes of the tangents drawn from P.

∴ m₁ + m₂ = `y_1/x_1`, m₁·m₂ = `"a"/x_1`

Since (x₁, y₁) and a are constants, m₁ m₂ is a constant.

`("m"_1/"m"_2)` = k, where k is a constant.