Question

$px+qy+\lambda (qx-py+r^{\prime }))=0$ and $px+qy-\lambda (qx-py+r^{\prime }))=0$ are two given lines. Determine the equations of their bisectors.

Answer 1

The required bisectors are
$\frac{P+\lambda Q}{\sqrt{\{{(p+\lambda q)}^2+{(q-\lambda p)}^2\}}}=\pm \frac{P-\lambda Q}{\sqrt{\{{(p-\lambda q)}^2+{(q+\lambda p)}^2\}}}$
The denominator on both sides cancel as each is $\sqrt{\left\{(p^2+q^2)(1+{\lambda }^2)\right\}}$
Hence the bisectors are $P+\lambda Q=\pm (P-\lambda Q)$
or $P=0$ or $Q=0$
$px+qy+r=0$ or $qx-py+r^{\prime }=0$
Clearly these lines are perpendicular as we know that the bisectors of angles of two given lines are always perpendicular.