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Question

Two sides of a rhombus ABCD are parallel to the lines y=x+2 and y=7x+3. If the diagonals of the rhombus intersect at the point (1,2) and the vertex A is on the Y-axis at a distance a unit from the origin, then a is equal to:

A
5
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B
2
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C
25
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D
52
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Solution

The correct option is D 52
Let the coordinate of A be (0,α).
Since the sides AB and AD are parallel to the lines y=x+2 and y=7x+3 respectively.
Therefore the diagonal AC is parallel to the bisector of the angle between these two lines.
The equation of the bisectors are given by,
xy+22=±7xy+3505(xy+2)=±(7xy+3)
2x+4y7=0 and 12x6y+13=0
Thus, the diagonals of the rhombus are parallel to the lines
2x+4y7=0 and 12x6y+14=0
Therefore slope of AE=24 or 126
2α10=12 or 2α10=2
α=52 or α=0

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