Two tangents are drawn to an ellipse 3x2 + 4y2 = 12 from a point P. If the points in which these tangents meet the coordinate axes are concylic, then the locus of point P is ax2 + by2 = 1. Find the value of (a + b).

Engineering

Mathematics

Standard Ellipse

Question

Two tangents are drawn to an ellipse 3x² + 4y² = 12 from a point P. If the points in which these tangents meet the coordinate axes are concylic, then the locus of point P is ax² + by² = 1. Find the value of (a + b).

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Let P (h, k). Equation of pair of tangents
(3x² + 4y² – 12) (3h² + 4k² – 12) = (3hx + 4ky – 12)²
Put y = 0
(3x² – 12) (3h² + 4k² – 12) = (3hx – 12)²
Product of roots OA · OB = $\frac{- 12 (3 h^{2} + 4 k^{2})}{3 (4 k^{2} - 12)}$
Put x = 0
(4y² – 12) (3h² + 4k² – 12) = (4ky – 12)²
Product of roots OC · OD = $\frac{- 12 (3 h^{2} + 4 k^{2})}{4 (3 h^{2} - 12)}$

Since A, B, C, D are concyclic
OA · OB = OC · OD
⇒ 4 (3h² – 12) = 3 (4k² – 12) ⇒ h² – 4 = k² – 3
Locus : h² – k² = 1. ⇒ Locus is x² – y² = 1
⇒ a + b = 0.

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