Question

If the tangent drawn from the point (–6, 9) to the parabola y² = kx are perpendicular to each other, find k

Answer 1

Given equation of the parabola is y² = kx.

Comparing this equation with y² = 4ax, we get

4a = k

∴ a = `"k"/4`

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

Since the tangent passes through the point (–6, 9),

9 = `- 6"m" + "k"/(4"m")`

∴ 36m = –24m² + k

∴ 24m² + 36m – k = 0

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

∴ m₁m₂ = `(-"k")/24`

Since the tangents are perpendicular to each other,

m₁m₂ = – 1

∴ `(-"k")/24` = – 1

∴ k = 24

Alternate method:

We know that, tangents drawn from a point on directrix are perpendicular.

∴ (–6, 9) lies on the directrix x = – a.

∴ – 6 = – a

∴ a = 6

Since 4a = k,

k = 4(6) = 24