The point, which is at the shortest distance from the line x + y = 7 and lying on an ellipse x 2 + 2 y 2 = 6 , has coordinates ( a , b ) then the value of a b is

Step-by-step solution

Equation of the ellipe is x 2 6 + y 2 3 = 1 The tangent at the point of shortest distance from the line x + y = 7 should be parallel to the given line Any point on the given ellipse is ( 6 cos ⁡ θ , 3 sin ⁡ θ ) Equation of the tangent is x cos ⁡ θ 6 + y sin ⁡ θ 3 = 1 . It is parallel to x + y = 7 ∵ Equation of the tangent at P ( a cos θ , b sin θ ) on ellipse is x cos θ a + y sin θ b = 1 ⇒ cos ⁡ θ 6 = sin ⁡ θ 3 ⇒ cos ⁡ θ 2 = sin ⁡ θ 1 ⇒ sin θ = 1 3 , cos θ = 2 3 The required point is ( 2 , 1 ) Hence, ( a , b ) = (