Question

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. $(\frac{1}{3},-\frac{8}{3})$
• D. $(-\frac{10}{3},-\frac{7}{3})$

Answer 1

The correct option is C

$(\frac{1}{3},-\frac{8}{3})$

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, $-1=\frac{x+1}{2}and-2=\frac{y+2}{2}$
$\Rightarrow x+1=-2andy=-4-2$
$\Rightarrow x=-3$
and $y=-6$
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 $(\frac{1}{3},\frac{-8}{3})$