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Maths-

General

Easy

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-

x₁ = y₂
x₁ = y₁
y₁ = y₂
x₂ = y₁

The correct answer is: y₁ = y₂

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)).
$therefore$ x₁ = at₁t₂and y₁ = a(t₁ + t₂)
$rightwards double arrow$ x₁ = –a and y₁ = a(t₁ + t₂) [ $because$ PQ is a focal chord, $therefore$ t₁t₂ = –1]
The normals at P and Q intersect at
(2a + a (t₁² + t₂² + t₁t₂), – at₁t₂ (t₁ + t₂))
$therefore$ x₂ = 2a + a (t₁² + t₂² + t₁t₂)
and y₂ = –at₁t₂ (t₁ + t₂)
$rightwards double arrow$ x₂ = 2a + a (t₁² + t₂² – 1) = a + a(t₁² + t₂²)
and y₂ = a (t₁ + t₂)
Clearly, y₁ = y₂.
Hence (C) is the correct answer.

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