Suppose AB is a focal chord of the parabola y²=12 x of length l and slope m<√3. If the distance of the chord AB from the origin is d, then l d² is equal to _________.

Solution

Maths sir · 4 steps

Write l and d in terms of the slope m, multiply them, and watch m disappear completely.

1. Read the parabola. y² = 12x is y² = 4ax with a = 3, so the focus is (3, 0). Every focal chord passes through this point.
2. Length of chord. A focal chord making angle θ with the x-axis has length 4a cosec²θ. With m = tan θ, cosec²θ = 1 + cot² θ = (1+m²)/m², so l = 12(1+m²)/m².
3. Distance from origin. The chord is y = m(x - 3), that is mx - y - 3m = 0. Its distance from (0, 0) is d = 3|m|/√(1+m²), so d² = 9m²/(1+m²).

   l d² = 12(1+m²)/m² × 9m²/(1+m²) = 108

4. So. The product does not depend on m, so the condition m < √3 does not change anything. l d² = 108.

The idea to keep. For y² = 4ax, a focal chord of slope m has length 4a(1+m²)/m² and distance a|m|/√(1+m²) from the vertex, so l d² = 4a³ every time.

Answer from the NTA final answer key. Solution by PrepWiser.