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a tangent to the ellipse x² + 4y²=4 meets the ellipse x²+2y²=6 at P and Q.the angle between the tangents at P and Q of the ellipse x² +2y²=6

$E q u a t i o n o f t h e e l l i p s e a r e, x^{2} + 4 y^{2} = 4 \Rightarrow \frac{x^{2}}{4} + \frac{y^{2}}{1} = 1 ...... (1) A n d, x^{2} + 2 y^{2} = 6 \Rightarrow \frac{x^{2}}{6} + \frac{y^{2}}{3} = 1 ...... (2) S i n c e \tan g e n t t o t h e e l l i p s e (1) m e e t s t h e e l l i p s e (2) a t P, Q, T h e r e f o r e, e q u a t i o n o f \tan g e n t o f e l l i p s e (1), \frac{x \cos θ}{2} + \frac{y \sin θ}{1} = 1 ..... (3) L e t i t c u t s e l l i p s e (2) a t p o int (h, k), t h e n i t i s c h o r d o f c o n t a c t o f (h, k) w i t h r e s p e c t t o e l l i p s e (2) . T h u s t h e e q u a t i o n i s, \frac{h x}{6} + \frac{k y}{3} = 1 ..... (4) S i n c e e q u a t i o n (3) a n d (4) r e p r e s e n t s t h e s a m e l i n e, S o, c o m p a r i n g (3) a n d (4), w e h a v e, \frac{\cos θ}{h / 3} = \frac{\sin θ}{k / 3} = 1 \Rightarrow \cos^{2} θ = \frac{h^{2}}{9} a n d \sin^{2} θ = \frac{k^{2}}{9} \Rightarrow \cos^{2} θ + \sin^{2} θ = \frac{h^{2}}{9} + \frac{k^{2}}{9} \Rightarrow \frac{h^{2}}{9} + \frac{k^{2}}{9} = 1 S o, l o c u s o f (h, k) i s x^{2} + y^{2} = 9 = 6 + 3 = a^{2} + b^{2} i . e ., d i r e c t o r c i r c l e o f e l l i p s e (2) H e n c e, t h e a n g l e b e t w e e n t h e \tan g e n t s i s 90^{0} .$

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