The normal to the parabola y = 2x^2 at the point P(1, 2) intersects the curve y = 2 x at P and another point Q . Let the area of the region bounded by this normal line and the curve y = 2 x be A = m n - p _e(2) , where m and n are coprime natural numbers. Then the value of m + n + p is equal to

Step-by-step solution

First, we find the equation of the normal to the parabola y = 2x^2 at the point P(1, 2) . Differentiating with respect to x : dy dx = 4x At x = 1 , the slope of the tangent is m_t = 4 . The slope of the normal is m_n = - 1 4 . The equation of the normal at P(1, 2) is: y - 2 = - 1 4 (x - 1) y = 9 4 - x 4 Next, we find the intersection of this normal with the curve y = 2 x : 9 4 - x 4 = 2 x 9x - x^2 = 8 x^2 - 9x + 8 = 0 (x - 1)(x - 8) = 0 The points of intersection are at x = 1 and x = 8 . The area A bounded by the n