Question

Tow consecutive sides of a parallelogram are $4x+5y=0$ and $7x+2y=0$.If the equation to one diagonal is $11x+7y=9$, then the equation of the other diagonal is

• A. $x+y=0$
• B. $2x+y=0$
• C. $x-y=0$
• D. None of these

Answer 1

Answer: (C)

$x-y=0$

We know that, Diagonals of a Parallelogram bisect each other and the given diagonal meets both the consecutive sides

$4x+5y=0\Rightarrow y=\frac{-4x}{5}$

$11x+7\left(\frac{-4x}{5}\right)=9\Rightarrow x=\frac{5}{3}\to y=\frac{-4x}{5}=-\frac{4}{3}$

Hence, one vertice of the parallelogram is $(\frac{5}{3},-\frac{4}{3})$

Similarly, $7x+2y=0\Rightarrow y=-\frac{7}{2}x$

$11x+7(-\frac{7}{2}x)=9\Rightarrow x=-\frac{2}{3}\to y=\frac{7}{3}$

Hence, opposite vertice of the parallelogram is $(-\frac{2}{3},\frac{7}{3})$

Now, the other diagonal must pass through the center of these obtained point of the given diagonal's end

Hence, the mid point of these point is $(\frac{\frac{5}{3}-\frac{2}{3}}{2},\frac{\frac{7}{3}-\frac{4}{3}}{2})\equiv (\frac{1}{2},\frac{1}{2})$

and the given adjacent sides are passing through the Origin $(0,0)$

So, that the origin is the third Vertice of the Parallelogram

hence, slope of other diagonal line $=\frac{\frac{1}{2}-0}{\frac{1}{2}-0}=1$

hence, the equation of the Other diagonal is

$x=y\Rightarrow x-y=0$