Question

The locus of the midpoint of the portion intercept between the axes by the line xcosa + ysina = P where P is a constant is ______.

Options

• x² + y² = 4p²

• `1/x^2 + 1/y^2 = 4/p^2`

• x² + y² = `4/p^2`

• `1/x^2 + 1/y^2 = 2/p^2`

Answer 1

The locus of the midpoint of the portion intercept between the axes by the line xcosa + ysina = P where P is a constant is `underlinebb(1/x^2 + 1/y^2 = 4/p^2)`.

Explanation:

Let mid point of AB is (h, k)

Mid point of AB is given by `(P/(2cosα), P/(2sinα))`

∴ h = `P/(2cosα)` and k = `P/(2sinα)`

⇒ cosα = `P/(2h)` and sinα = `P/(2k)`

On squaring and adding, we get

cos²α + sin²α = `P^2/(4h^2) + P^2/(4k^2)`

⇒ 1 = `P^2/(4h^2) + P^2/(4k^2)`

⇒ `4/P^2 = 1/h^2 + 1/k^2`

∴ Locus is ⇒ `4/p^2 = 1/x^2 + 1/y^2`