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Straight Line — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsStraight Line2 Marks
Question
Prove the Theorem : Non-vertical lines having slopes $m_1$ and $m_2$ are perpendicular to each other if and only if $m_1 \times m_2=-1$.
✓

Answer

Let $\alpha$ and $\beta$ be inclinations of lines having slopes $m_1$ and $m_2$. As lines are non vertical $\alpha \neq \frac{\pi}{2}$ and $\beta \neq \frac{\pi}{2}$
$
\therefore \tan \alpha=m_1 \text { and } \tan \beta=m_2
$
From Fig. 5.5 and 5.6 we have,
$\begin{aligned} & \alpha-\beta=90^{\circ} \text { or } \alpha-\beta=-90^{\circ} \\ & \alpha-\beta= \pm 90^{\circ} \\ & \therefore \cos (\alpha-\beta)=0 \\ & \therefore \cos \alpha \cos \beta+\sin \alpha \sin \beta=0 \\ & \therefore \sin \alpha \sin \beta=-\cos \alpha \cos \beta \\ & \therefore \tan \alpha \tan \beta=-1 \\ & \therefore m_1 m_2=-1\end{aligned}$

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