The area of the region bounded by the curve y = x^3 and its tangent at a point P(h, k) in the first quadrant is given as 108 . The value of h + k is

Step-by-step solution

Let the point P be (a, a^3) . Since P lies in the first quadrant, a > 0 . The given curve is y = x^3 . Differentiating with respect to x , we get dy dx = 3x^2 . The slope of the tangent at P(a, a^3) is 3a^2 . The equation of the tangent at P is y - a^3 = 3a^2(x - a) , which simplifies to y = 3a^2x - 2a^3 . To find the points of intersection of the curve and the tangent, we equate them: x^3 = 3a^2x - 2a^3 x^3 - 3a^2x + 2a^3 = 0 (x - a)^2(x + 2a) = 0 The tangent intersects the curve again at x = -2a . The area of the