If tangent drawn at a point t 2 , 2 t on the parabola y 2 = 4 x is same as the normal drawn at a point 5 cos ϕ , 2 sin ϕ on the ellipse 4 x 2 + 5 y 2 = 20 . Find the values of t & ϕ .

Equation of tangent at x 1 , y 1 for parabola y 2 = 4 x is y y 1 = 4 x + x 1 2 . Therefore, Equation of tangent at t 2 , 2 t is 2 t y = 2 x + t 2 . ⇒ x - t y + t 2 = 0 … 1 Now, Equation of normal to an ellipse x 2 a 2 + y 2 b 2 = 1 at a cos θ , b sin θ is a x sec θ - b y cosec θ = a 2 - b 2 Therefore, for ellipse x 2 5 + y 2 4 = 1 at 5 cos ϕ , 2 sin ϕ is 5 sec ϕ x - 2 c osec ϕ y = 5 - 4 ⇒ 5 sec ϕ x - 2 cosec ϕ y = 1 … 2 Given that Equations 1 and 2 are same line then ratios of their co-efficient is same, 5 sec ϕ 1 = - 2 cosec ϕ - t = - 1 t 2 ⇒ t = 2 5 cot ϕ and t = - 1 2 sin ϕ ⇒ 2 5 cot ϕ = - 1 2 sin ϕ ⇒ 4 cos ϕ = - 5 sin 2 ϕ ⇒ 5 cos 2 ϕ - 4 cos ϕ - 5 = 0 ⇒ cos ϕ - 5 5 cos ϕ + 1 = 0 ⇒ cos ϕ = - 1 5 (since cosϕ ≠ 5 ) ⇒ ϕ = cos - 1 - 1 5 Now, Substituting value of cos ϕ in t = - 1 2 sin ϕ we get, t = - 1 2 1 - 1 5 ⇒ t = ± 1 5 Hence, t = ± 1 5 and ϕ = cos - 1 - 1 5 .