The area of the region bounded by the parabola ( y - 2 ) 2 = ( x - 1 ) , the tangent to it at the point whose ordinate is 3 and the x -axis, is:

Step-by-step solution

Given, the parabola y - 2 2 = x - 1 and y = 3 , x = 2 So, point is ( 2 , 3 ) Differentiate given equation w.r.t. x , we get 2 ( y - 2 ) y ' = 1 y ′ = 1 2 ( y − 2 ) y ′ ( 2 , 3 ) = 1 2 So, the equation of tangent at 2 , 3 y - 3 = 1 2 x - 2 x - 2 y + 4 = 0 Required area = ∫ 0 3 [ ( y - 2 ) 2 + 1 ] d y - ∫ 0 3 ( 2 y - 4 ) d y Required area = ∫ 0 3 y 2 - 4 y + 5 d y + ∫ 0 3 2 y - 4 d y Required area = y 3 3 - 2 y 2 + 5 y 0 3 - y 2 - 4 y 0 3 Required area = 9   sq units