Let RS be the diameter of the circle x 2 + y 2 = 1 where, S is the point ( 1 , 0 ) . Let P be a variable point (other than R & S ) on the circle and tangents to the circle at S & P meet at the point Q . The normal to the circle at P intersects a line drawn through Q parallel to R S at point E . Then, the locus of E passes through the point (s):

Step-by-step solution

Let P be ( cos θ , sin θ ) , where θ ≠ 0 , π Tangent at P : x cos ⁡ θ + y sin ⁡ θ = 1 ........(i) Tangent at S : x = 1 .........(ii) ∴ By (i) and (ii) : Q 1 , 1 - cos ⁡ θ sin ⁡ θ Line through Q parallel to RS : y = 1 - cos ⁡ θ sin ⁡ θ ⇒ y = tan ⁡ θ 2 ..........(iii) Normal at P : y = sin ⁡ θ cos ⁡ θ x ⇒ y = tan ⁡ θ . x ⇒ y = 2 t an θ 2 1 - t an 2 θ 2 x ........(iv) Point of intersection of equation (iii) and (iv), E : h = 1 - tan 2 ⁡ θ 2 2 ; k = tan ⁡ θ 2 Eliminating θ : h = 1 - k 2 2 ⇒ y 2 = 1 - 2 x ( 1 3 , 1 3 )