Let C be the circle of minimum area touching the parabola y = x^2 + 18 and the lines 12|x| - 5y = 0 . Let C₀ be the center of this circle. If a tangent is drawn from the origin O(0,0) to the circle C touching it at point P , then the area of the triangle OPC₀ is

Step-by-step solution

The vertex of the parabola y = x^2 + 18 is (0, 18) . By symmetry, the center of the circle C lies on the y-axis. Let the center be C₀(0, c) . Since the circle touches the parabola at its vertex and is of minimum area, it lies below the vertex. Thus, its radius is r = 18 - c , where c The circle also touches the lines 12x - 5y = 0 and 12x + 5y = 0 . The perpendicular distance from the center (0, c) to the line 12x - 5y = 0 must be equal to the radius r . |12(0) - 5(c)| 12^2 + (-5)^2 = r 5c 13 = 18 - c 5c = 234 - 13c