Let T be the tangent to the ellipse E : x 2 + 4 y 2 = 5 at the point P ( 1 , 1 ) . If the area of the region bounded by the tangent T , ellipse E , lines x = 1 and x = 5 is α 5 + β + γ cos - 1 1 5 , then | α + β + γ | is equal to______.

Step-by-step solution

We have, x 2 + 4 y 2 = 5 ⇒ x 2 5 + y 2 5 4 = 1 Tangent at P 1 , 1 to the ellipse is x + 4 y = 5 Required Area = ∫ 1 5 5 - x 4 - 5 - x 2 2 d x = 5 x 4 - x 2 8 - x 4 5 - x 2 - 5 4 sin - 1 x 5 1 5 = 5 5 4 - 5 8 - 5 4 · π 2 - 5 4 + 1 8 + 1 2 + 5 4 sin - 1 1 5 = 5 5 4 - 5 4 - 5 4 · π 2 + 5 4 π 2 - cos - 1 1 5 = 5 5 4 - 5 4 - 5 π 8 + 5 π 8 - 5 4 cos - 1 1 5 = 5 4 5 - 5 4 - 5 4 cos - 1 1 5 So, α = 5 4 ,   β = - 5 4   &   γ = - 5 4 α + &#