A tangent having slope - 4 3 to the ellipse x 2 18 + y 2 32 = 1 intersects the major and minor axes at A and B . If O is the origin, then the area of Δ O A B is

Step-by-step solution

Any point on the ellipse x 2 3 2 2 + y 2 4 2 2 = 1 can be taken as 3 2 cos ⁡ θ , 4 2 sin ⁡ θ and the slope of the tangent as - b 2 x a 2 y Hence, slope = - 32 3 2 cos ⁡ θ 18 4 2 sin ⁡ θ = - 4 3 cot ⁡ θ … . 1 Given, the slope of the tangent = - 4 3 … . 2 From equations (1) and (2), we get, cot ⁡ θ = 1 ⇒ θ = π 4 Hence, the equation of the tangent is x . 1 2 3 2 + y . 1 2 4 2 = 1 (i.e.) x 6 + y 8 = 1 Hence, A = 6 , 0 , B = 0 , 8 Area of Δ O A B = 1 2 × 6 × 8 = 24 s q . u n i t s