Question

Answer the following:

Show that the two tangents drawn to the parabola y² = 24x from the point (−6, 9) are at the right angle

Answer 1

Given equation of the parabola is y² = 24x

Comparing this equation with y² = 4ax, we get

4a = 24

∴ a = `24/4` = 6

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

∴ y = `"m"x + 6/"m"`

But, (– 6, 9) lies on the tangent

∴ 9 = `-6"m" + 6/"m"`

∴ 9m = – 6m² + 6

∴ 6m² + 9m – 6 = 0

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

∴ m₁m₂ = `(-6)/6` = – 1

∴ Tangents drawn to the parabola y² = 24x from the point (– 6, 9) are at right angle.

Alternate method:

Comparing the given equation with y² = 4ax, we get

4a = 24

∴ a = 6

Equation of the directrix is x = – 6.

The given point lies on the directrix.

Since tangents are drawn from a point on the directrix are perpendicular,

Tangents drawn to the parabola y² = 24x from the point (– 6, 9) are at the right angle.