The sides of a rhombus ABCD are parallel to the lines, x-y+2=0 and 7x-y+3=0. If the diagonals of the rhombus intersect at P(1,2) and the vertex A (different from the origin) is on the y-axis, then the ordinate of A is?

Let the coordinate A be (0,c).
Equations of the parallel lines are given: x-y+2=0 and 7x-y+3=0
We know that the diagonals will be parallel to the angle bisectors of the two sides y=x+2 and y=7x+3
x-y+2/√2 = ±(7x-y+3)/√50
5x - y + 10 = ±(7x - y + 3)
Parallel equations of the diagonals are 2x+4y-13=0 and 12x-8y+7=0
Slopes of diagonals m = -1/2 and 2
We know that the slope of diagonal from A(0,c) and passing through P(1,2) is (2-c)/1
Therefore, (2-c) = -1/2 => c = 5/2
Or (2-c) = 2 => c = 0, but it is given that A is not the origin, so
2-c = -1/2 => c = 5/2
Therefore, the coordinate of A is (0, 5/2)