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in Mathematics by (53.1k points)

The sides of a rhombus are parallel to the lines y + 2x = 3 and y + 7x = 2. The diagonals intersect at (1, 2). If one vertex lies on the y-axis, then find the coordinates of this vertex

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See Fig.  Let ABCD be the rhombus where A = (0, k). AB and CD are parallel to y + 7x + 2 whereas BC and AD are parallel to y = 2x + 3. Hence, the equation of AB is\

y = 7x + k   ...(1)

Since (1, 2) is the midpoint of AC and A = (0, k), it follows that C = (2, 4 − k). Also, BC is parallel to 2x + 3 and passes through C (2, 4 − k). Hence, the equation of BC is

y -(4 - k) = 2(x - 2)

y = 2x - k  ...(2)

 Solving Eqs. (1) and (2), we have

B = (-2k/5, -9k/5)

 Since A = (0,k), B = (-2k/5, - 9k/5), C = (2,4 - k) and AB = BC,we have

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