The focus and corresponding directrix of an ellipse are 3,4 and x + y - 1 = 0 respectively. If the eccentricity of the ellipse is 1 2 , then the coordinates of the centre of the ellipse are

Step-by-step solution

Let, the lengths of the semi-major axis and semi-minor axis of the ellipse are a & b respectively, then the distance of the focus 3,4 from the corresponding directrix x + y - 1 = 0 is equal to a e - a e ⇒ a 1 2 - a 1 2 = 3 + 4 - 1 2 = 3 2 ⇒ 3 a 2 = 3 2 ⇒ a = 2 2 Distance between the focus and centre = a e = 2 The slope of the axis of the ellipse is 1 = tan ⁡ θ ⇒ cos ⁡ θ , sin ⁡ θ = 1 2 , 1 2 or - 1 2 , - 1 2 The points on the axis of the ellipse at a distance 2 units from 3,4 are 3 ± 2 1 2 , 4 ± 2 1 2 ⇒ Points are