Question

Find the locus of the intersection of tangents to the parabola y² = 4ax, the angle between them being always a given angle α.

Answer 1

The straight line y = mx + a/m is always a tangent to the parabola.

If it pass through the point T(h, k) we have

m²h - mk + a = 0 ......(1)

If m₁ and m₂ be the roots of this equation we have

and the equations to TP and TQ are then

Hence the coordinates of the point T always satisfy the equation

We shall find in a later chapter that this curve is a hyperbola.

As a particular case let the tangents intersect at right angles, so that m₁m₂ = - 1.

From (3) we then have h = - a, so that in this case the point T lies on the straight line x = - a, which is the directrix.

Hence the locus of the point of intersection of tangents, which cut at right angles, is the directrix.