The area of the region bounded by the parabola y = 3x^2 + 12 and the two tangents drawn to it from the origin is equal to

Step-by-step solution

Let the equation of a tangent from the origin (0,0) be y = mx . To find the points of tangency, substitute y = mx into the equation of the parabola: 3x^2 + 12 = mx 3x^2 - mx + 12 = 0 For the line to be a tangent, the roots of this quadratic must be equal, so its discriminant must be zero: = m^2 - 4(3)(12) = 0 m^2 - 144 = 0 m = 12 Thus, the two tangents are y = 12x and y = -12x . Substituting m = 12 into the quadratic gives 3x^2 - 12x + 12 = 0 3(x-2)^2 = 0 x = 2 . Substituting m = -12 gives 3x^2 + 12x + 12 = 0 3(x+2