Problem 4 - Entrance Test

Let P divide the line segment joining A(-3, -4) and B(1, -2) in the ratio k : 1. Then the coordinates of P are ((k*1 + 1*(-3))/(k + 1), (k*(-2) + 1*(-4))/(k + 1)) = ((k - 3)/(k + 1), (-2k - 4)/(k + 1)). Similarly, the coordinates of Q are ((1*k + 1*1)/(k + 1), (1*(-2) + 1*(-4))/(k + 1)) = ((k + 1)/(k + 1), (-2 - 4)/(k + 1)) = (1, -6/(k + 1)). It is given that the coordinates of P are (x, y) and that of Q are (y, x). So we get, (k - 3)/(k + 1) = y and (-2k - 4)/(k + 1) = x. From the second equation, k = (-xk - x - 4)/(x + 2). Substituting the value of k in the first equation, we get ((-xk - x - 4)/(x + 2) - 3)/((-xk - x - 4)/(x + 2) + 1) = y. Simplifying gives ((-xk - x - 4 - 3x - 6)/(x + 2))/((-xk - x - 4 + x + 2)/(x + 2)) = y. This simplifies to (-4x - x - 10)/(-x - 2) = y, which gives x = -1. Putting the value of x in (-2k - 4)/(k + 1) = x, we get (-2k - 4)/(k + 1) = -1. Solving gives -2k - 4 = -k - 1, so -k = 3 and k = -3. So (k - 3)/(k + 1) = (-3 - 3)/(-3 + 1) = -6/-2 = -3. Hence the coordinates of P and Q are (-1, -3) and (0, -2) respectively but since none of the options contain (-1, -3), (-3, -1) we can check if the point is (0, -3), (-2, 0).