Line Equally Inclined with two lines - Practice Questions & MCQ

Line Equally Inclined with two lines

If the two lines with slope $m_1$ and $m_2$ are equally inclined to a line having slope $m$, then

$
\left(\frac{\mathbf{m}_1-\mathbf{m}}{1+\mathbf{m}_1 \mathbf{m}}\right)=-\left(\frac{\mathbf{m}_2-\mathbf{m}}{1+\mathbf{m}_2 \mathbf{m}}\right)
$
Two lines with slope $m_1$ and $m_2$ intersect at point $A$

As from the fig

$
\begin{aligned}
& \angle \mathrm{PAQ}=\angle \mathrm{QAR}=\angle \theta \\
& \tan (\angle \mathrm{PAQ})=\frac{\mathrm{m}_1-\mathrm{m}}{1+\mathrm{m}_1 \mathrm{~m}}=\tan \theta \\
& \text { and } \\
& \tan (\angle \mathrm{QAR})=\frac{\mathrm{m}-\mathrm{m}_2}{1+\mathrm{mm}_2}=\tan \theta
\end{aligned}
$
Hence,

$
\left(\frac{\mathbf{m}_1-\mathbf{m}}{1+\mathbf{m}_1 \mathbf{m}}\right)=-\left(\frac{\mathbf{m}_2-\mathbf{m}}{1+\mathbf{m}_2 \mathbf{m}}\right)
$