The area of the region bounded by the parabola y^2 = 4x and the line y = 2x - k is 9 square units. The value of the positive constant k is

Step-by-step solution

Given curves are y^2 = 4x and y = 2x - k . Expressing x in terms of y , we have x = y^2 4 and x = y+k 2 . To find the points of intersection, equate the values of x : y^2 4 = y+k 2 y^2 - 2y - 2k = 0 Let the roots of this quadratic equation be y₁ and y₂ (with y₂ > y₁ ). The difference between the roots is: y₂ - y₁ = (-2)^2 - 4(1)(-2k) = 2 1+2k The area of the bounded region is given by: A = _ y₁ ^ y₂ ( y+k 2 - y^2 4 ) dy Using the standard result _ ^ (ax^2+bx+c) dx = |a| 6 ( - )^3 for roots , : A = 1/4 6 (y₂ - y₁)^3