Question

If the tangents and normals at the extremities of a focal chord of a parabola intersect at (x₁, y₁) and (x₂, y₂) respectively, then-

• A. x₁ = y₂
• B. x₁ = y₁
• C. y₁ = y₂
• D. x₂ = y₁

Answer 1

Answer: (C)

y₁ = y₂

Answer 2

Let P(at₁², 2at₁) and Q(at₂², at₂) be the extremities of a focal chord of the parabola y² = 4ax. The tangents at P and Q intersect at (at₁t₂, a(t₁ + t₂)).

\ x₁ = at₁t₂ and y₁ = a(t₁ + t₂)

̃ x₁ = –a and y₁ = a(t₁ + t₂)

[Q PQ is a focal chord, \ t₁t₂ = –1]

The normals at P and Q intersect at

(2a + a (t₁² + t₂² + t₁t₂), – at₁t₂ (t₁ + t₂))

\ x₂ = 2a + a (t₁² + t₂² + t₁t₂)

and y₂ = –at₁t₂ (t₁ + t₂)

̃ x₂ = 2a + a (t₁² + t₂² – 1) = a + a(t₁² + t₂²)

and y₂ = a (t₁ + t₂)

Clearly, y₁ = y₂.

Hence (3) is the correct answer.