Question:

Let two intersecting lines OA and OB, intersect at origin O and let both lines OA and OB makes equal angles with x axis.<br>i.e., <br>∠ XOA = ∠ XOB = θ.<br>∴ Equations of straight lines OA and OB are<br>y = x tanθ and y = -x tanθ<br>or x sinθ - y cosθ = 0 .......... (i)<br>and x sinθ + y cosθ = 0 .......... (ii)<br>Let P(α, β) is the point whose locus is to be determine.<br>According to the example (PM)² + (PN)² = 2λ² (say)<br>∴ (α sinθ + β cosθ)² + (α sinθ - β cosθ)² = 2λ² ⇒ 2α² sin²θ + 2β² cos²θ = 2λ²<br>or α² sin²θ + β² cos²θ = λ²⇒ α²/λ² cosec²θ + β²/λ² sec²θ = 1<br>Hence required locus is x²/(λ cosec θ)² + y²/(λ sec θ)² = 1