Question

The following four lines $ax+by+c=0$ and
$ax+by+d=0,a^{\prime }x+b^{\prime }y+c^{\prime }=0,a^{\prime }x+b^{\prime }y+d^{\prime }=0$
represent in pairs the sets of parallel lines of a
parallelogram. Write the equation of its diagonals.

Answer 1

The diagonal $AC$ can be written
$P+\lambda Q=0$ i.e. $\left(throughA\right)$
$(a,b,c)+\lambda (a,b,d)=0$
It can also be written as $\left(throughC\right)$
$(a,b,d)+\mu \left(a^{\prime }{b}^{\prime }{c}^{\prime }\right)=0$
Comparing the co-efficients of $x,y$ and constant in $\left(I\right)$ and $\left(II\right)$ $\lambda \mu$ and $c+\lambda d=d+\mu c$
Or $c-d=\lambda (c-d)$ $\therefore \lambda =\frac{c-d}{c^{\prime }-d^{\prime }}=\mu$
$\therefore ACis(c^{\prime }-d^{\prime })(ax+by+c)+(c-d)(a^{\prime }x+b^{\prime }y+d^{\prime })=0$