The line 2 x + y = 3 cuts the ellipse 4 x 2 + y 2 = 5 at P and Q . If θ is the acute angle between normal at P and tangent at Q , then θ is equal to

Step-by-step solution

Solving y = 3 - 2 x and 4 x 2 + y 2 = 5 , we get the intersection point P 1 2 , 2 and Q 1 , 1 The equation of tangent at these points are 4 x 1 2 + 2 y = 5 and 4 x 1 + y 1 = 5 ⇒ Slope of tangents at P and Q are - 1 and - 4 respectively ⇒ Slope of normal at P is 1 ⇒ tan ⁡ θ = 1 - - 4 1 + 1 - 4 = 5 3 ⇒ cos ⁡ θ = 3 34