Let C be the circle of minimum area touching the parabola x = y^2 + 8 and the lines 3x - 4|y| = 0 . The length of the intercept made by the circle C on the x-axis is

Step-by-step solution

The vertex of the parabola x = y^2 + 8 is (8, 0) . By symmetry, the center of the circle C lies on the x-axis. Let the center be (c, 0) . Since the circle touches the parabola at its vertex and is of minimum area, it lies to the left of the vertex. Thus, its radius is r = 8 - c , where c The circle also touches the lines 3x - 4y = 0 and 3x + 4y = 0 . The perpendicular distance from the center (c, 0) to the line 3x - 4y = 0 must be equal to the radius r . |3(c) - 4(0)| 3^2 + (-4)^2 = r 3c 5 = 8 - c 3c = 40 - 5c 8c =