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Straight Line — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsStraight Line2 Marks
MCQ
A straight line through origin bisects the line passing through the given points $(a \cos \alpha, a \sin \alpha)$ and (a $\cos \beta$, a $\sin \beta)$, then the lines are
  • A
    Perpendicular
  • B
    Parallel
  • C
    Angle between them is $\frac{\pi}{4}$
  • D
    None of these
✓

Answer

(A)
Mid point of (a $\cos \alpha$, a $\sin \alpha$ ) and (a $\cos \beta, a \sin \beta$ ) is
$P \left(\frac{ a (\cos \alpha+\cos \beta)}{2}, \frac{ a (\sin \alpha+\sin \beta)}{2}\right)$
Image
∴ Slope of line $A B$ is
$\frac{a \sin \beta-a \sin \alpha}{a \cos \beta-a \cos \alpha}=\frac{\sin \beta-\sin \alpha}{\cos \beta-\cos \alpha}=m_1$
and slope of $O P$ is $\frac{\sin \alpha+\sin \beta}{\cos \alpha+\cos \beta}= m _2$
Now, $m_1 \times m_2=\frac{\sin ^2 \beta-\sin ^2 \alpha}{\cos ^2 \beta-\cos ^2 \alpha}=-1$
Hence, the lines are perpendicular.

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