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Two sides of a rhombus are along the lin...

Two sides of a rhombus are along the lines x-y+1=0 and 7x-y-5=0. If its diagonals intersect at (-1, -2), then which one of the following is a vertex of this rhombus?

A

`(-3,-9)`

B

`(-3,-8)`

C

`((1)/(3),-(8)/(3))`

D

`(-(10)/(3),-(7)/(3))`

Text Solution

Verified by Experts

As the given lines `x-y+1=0` and `7x-y-5=0` are not parallel, therefore they represent the adjacent sides of the rhombus.
On solving `x-y+1=0` and `7x-y-5=0`, we get `x=1` and `y=2`. Thus, one of the vertex is `A(1,2)`.

Let the coordinate of point `C` be `(x,y)`.
Then, `-1=(x+1)/(2)` and `-2=(y+2)/(2)`
`impliesx+1=-2` and `y=-4-2`
`impliesx=-3`
`impliesy=-6`
Hence, coordinates of `C=(-3,-6)`
Note that, vertices `B` and `D` will satisfy `x-y+1=0` and `7x-y-5=0`, respectively.
Since, option `(c )` satisfies `7x-y-5=0`, therefore coordinate of vertex `D` is `((1)/(3),(-8)/(3))`.
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