A point P(x, y) moves such that its distance from the point (4, 0) is twice its distance from the line x = 1 . If the locus of P intersects the line x = 3 at two distinct points A and B , then the value of AB^2 is equal to

Step-by-step solution

Let the point be P(x, y) . The distance of P from the point S(4, 0) is SP = (x - 4)^2 + y^2 . The distance of P from the line x = 1 is PM = |x - 1| . According to the given condition, SP = 2PM . Squaring both sides, we get SP^2 = 4PM^2 . (x - 4)^2 + y^2 = 4(x - 1)^2 x^2 - 8x + 16 + y^2 = 4(x^2 - 2x + 1) x^2 - 8x + 16 + y^2 = 4x^2 - 8x + 4 3x^2 - y^2 = 12 This is the equation of the locus of P , which represents a hyperbola. To find the points of intersection with the line x = 3 , substitute x = 3 into the locus equ