Let PQ be a focal chord of the parabola y^2 = 4ax ( a > 0 ) making an acute angle with the positive x -axis. Let the ordinate of P be positive and M be a point on the line segment PQ such that PM : MQ = 3 : 1 . If the line passing through M and perpendicular to the line PQ has the equation 4x + 3y = 111 , then the length of the latus rectum of the parabola is

Step-by-step solution

The equation of the line perpendicular to PQ is 4x + 3y = 111 , which has a slope of - 4 3 . Therefore, the slope of the focal chord PQ is 3 4 . For the parabola y^2 = 4ax , the endpoints of a focal chord can be taken as P(at^2, 2at) and Q ( a t^2 , - 2a t ) . The slope of the focal chord PQ is given by 2 t - 1 t . Equating this to 3 4 , we get: 2 t - 1 t = 3 4 t - 1 t = 8 3 3t^2 - 8t - 3 = 0 (3t + 1)(t - 3) = 0 Since the ordinate of P is positive, 2at > 0 t > 0 . Thus, t = 3 . Substituting t = 3 , the coordinates