Let x = 4 be a directrix to an ellipse whose centre is at the origin and its eccentricity is 1 2 . If P ( 1 , β ) , β > 0 is a point on this ellipse, then the equation of the normal to it at P is

Step-by-step solution

a e = 4 ⇒ a = 4 e ⇒ a = 2 b 2 = a 2 1 - e 2 = 3 ( 1 , β )  lies on  x 2 4 + y 2 3 = 1 ⇒ 1 4 + β 2 3 = 1 ⇒        β 2 = 9 4 ⇒ β = 3 2 ( ∵ β > 0 ) Normal at ( 1 , β ) ⇒   a 2 x 1 - b 2 y β = a 2 - b 2     ⇒ 4 x - 3 y β = 1 so equation of normal is 4 x - 2 y = 1