Question

Show that the locus of the point from which the pair of tangents drawn to the parabola y² = 4ax including constant angle α is cot²α(y² - 4ax) = (x + a)² .

Answer 1

Let P(x₁, y₁) be the point of intersection of tangents drawn at Q(at₁², 2at₁) and R(at₂², 2at₂) so that

x₁ = at₁t₂   .....(1)

and   Y₁ = a(t₁ + t₂)  ...(2)

It is known that 1/t₁  and 1/t₂ are the slopes of the tangent at Q and R so that by hypothesis, we have

Therefore, from Eqs. (1) and (2), we get

Therefore, the locus of (x₁, y₁) is

(x + a)² = cot²α(y² - 4αx)