Question

Two consecutive sides of a parallellogram are $4x+5y=0$ & $7x+2y=0$. If the equation to one diagonal is $11x+7y=9$ , find the equation to the other diagonal.

Answer 1

We know that Diagonals of parallelogram bisect each other

Now we will find the point of intersection of the diagonal and adjacent sides

Solving Simultaneously $4x+5y=011x+7y=9x_1=\frac{5}{3}{y}_1=\frac{-4}{3}$

and $7x+2y=011x+7y=9x_2=\frac{-2}{3}{y}_2=\frac{7}{3}$

Their midpoint will be $(\frac{\frac{5-2}{3}}{2},\frac{\frac{7-4}{3}}{2})=(\frac{1}{2},\frac{1}{2})$

Slope of the diagonal will be $\frac{\frac{1}{2}-0}{\frac{1}{2}-0}=1$ (as adjacent sides meet on origin so origin is also the vertex of parallelogram)

Now, Equation of diagonal will be $y-\frac{1}{2}=1(x-\frac{1}{2})\Rightarrow y-x=0$