A focal chord of the parabola x^2 - 10x - 12y + 37 = 0 has a positive slope and its length is 60 . The x -intercept of this focal chord is equal to _____ .

Step-by-step solution

The given equation of the parabola is x^2 - 10x - 12y + 37 = 0 . Completing the square, we get: x^2 - 10x + 25 = 12y - 12 (x-5)^2 = 12(y-1) Comparing this with the standard form (x-h)^2 = 4a(y-k) , we have 4a = 12 a = 3 . The vertex is (5, 1) and the focus is (h, k+a) = (5, 4) . The length of a focal chord for a vertical parabola with slope m is given by L = 4a(1+m^2) . Given that the length is 60 , we have: 12(1+m^2) = 60 1+m^2 = 5 m^2 = 4 Since the focal chord has a positive slope, m = 2 . The equation of the foc