Question

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

Answer 1

AB is $4x+5y=0.$
AD is $7x+2y=0.$

Both of them intersect at $A(0,0).$

Then diagonal $11x+7y=9$ does not pass through A i.e., $(0,0)$ and hence it cannot be the equation of AC. Therefore it represents diagonal BD.

Solving the diagonal BD with AB and AD respectively, we get the points B and D as $(5/3,4/3),(-2/3,7/3).$

The mid-point P of diagonal BD is $(1/2,1/2).$ Since the diagonals of a parallelogram bisect each other therefore diagonal AC will pass through

A and P, i.e., $(0,0)$ and $(\frac{1}{2},\frac{1}{2}).$

Its equation is clearly $y=x$ or $x-y=0$.