Question

The locus of the middle points of chords of the parabola y² = 4ax, which are of constant length 2l is-

• A. (4x + y²) (y² – 4) = 4l²
• B. (4y + x²) (x² – 4) = 4l²
• C. (4x – y²) (y² + 4) = 4l²
• D. None of these

Answer 1

Answer: (C)

(4x – y²) (y² + 4) = 4l²

Answer 2

Let R º (h, k) be mid-point on locus

P º (x₁, y₁) and Q º (x₂, y₂) be points on parabola

forming a chord so 2h = x₁ + x₂, 2k = y₁ + y₂

we have (x₁ – x₂)² + (y₁– y₂)² = (2l)²

also P, Q lies on parabola ̃ y₁² = 4x₁, y₂² = 4x₂

so $\left( \frac{y_{1}^{2}}{4} - \frac{y_{2}^{2}}{4} \right)$+ (y₁ – y₂)² = 4l²

̃ (y₁ – y₂)² (k² + 4) = 16l²

2y₁y₂ = 4k² – 8h ̃ (y₁– y₂)² = 16h – 4k²

So we have (16h – 4k²) (k² + 4) = 16l²

So locus (4x – y²) (y² + 4) = 4l²