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In a parallelogram ABCD , the equations of the adjacent sides AB and AD are x - y = -1 and x + 3y = 7 respectively. The point of intersection of the diagonals is M(4, a) . If the vertex C lies on the line x + 2y = 5a , then the value of BD^2 is equal to

Correct answer

16

Step-by-step solution

Let the vertices of the parallelogram be A, B, C, and D . The coordinates of A can be found by solving the equations of AB and AD : x - y = -1 x + 3y = 7 Subtracting the first from the second gives 4y = 8 y = 2 . Then x = 1 . So, A (1, 2) . In a parallelogram, the diagonals bisect each other, so M(4, a) is the midpoint of AC . Let C (x_c, y_c) . Then: 1 + x_c 2 = 4 x_c = 7 2 + y_c 2 = a y_c = 2a - 2 So, C (7, 2a - 2) . Since C lies on the line x + 2y = 5a , we substitute its coordinates: 7 + 2(2a - 2) = 5a 7 + 4a -

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