Question

AB, AC are tangents to a parabola y² = 4ax. If l₁, , l₃ are the lengths of perpendiculars from A, B, C on any tangent to the parabola, then –

• A. l₁, , l₃ are in G.P.
• B. , l₁, l₃ are in G.P.
• C. l₁, l₃, are in G.P.
• D. l₃, , l₁ are in G.P.

Answer 1

Answer: (B)

, l₁, l₃ are in G.P.

Answer 2

Let the coordinates of B and C be (at₁², 2at₁) and (at₂², 2at₂) respectively. Then, the coordinates of A are (at­₁t₂ , a(t₁ + t₂)).

The equation of any tangent to y² = 4ax is

ty = x + at²

\ l₁ = $\frac{at_{1}t_{2} - a(t_{1} + t_{2})t + at^{2}}{\sqrt{1 + t^{2}}}$,

l₂ = $\frac{at_{1}^{2} - 2att_{1} + at^{2}}{\sqrt{1 + t^{2}}}$

and l₃ = $\frac{at_{1}^{2} - 2att_{2} + at^{2}}{\sqrt{1 + t^{2}}}$

Clearly, l ₂ l ₃ = l₁². Therefore, l ₂, l ₁, l ₃ are in G.P.

Hence (2) is the correct answer.