A normal to the curve y = x^2 + 1 at a point P in the first quadrant intersects the x-axis at a point Q . If the area of the triangle OPQ (where O is the origin) is 5 sq. units, then the square of the length of the line segment PQ is equal to _____.

Step-by-step solution

Let the point P on the curve be (h, h^2+1) with h > 0 . The derivative of the curve is dy dx = 2x . The slope of the normal at P is m = - 1 2h . The equation of the normal at P is: y - (h^2+1) = - 1 2h (x - h) To find the x-intercept Q , we set y = 0 : -(h^2+1) = - 1 2h (x - h) 2h(h^2+1) = x - h x = 2h^3 + 3h So, the coordinates of Q are (2h^3+3h, 0) . The area of OPQ is given by: Area = 1 2 base height = 1 2 (2h^3+3h)(h^2+1) = 5 (2h^3+3h)(h^2+1) = 10 2h^5 + 5h^3 + 3h - 10 = 0 By inspection, h = 1 is the unique pos