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STD 11 - 9. straight line — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsSTD 11 - 9. straight line1 Mark
MCQ
A straight line through origin bisect the line passing through the given points $(a\cos \alpha ,a\sin \alpha )$ and $(a\cos \beta ,a\sin \beta )$, then the lines are
  • ✓
    Perpendicular
  • B
    Parallel
  • C
    Angle between them is $\frac{\pi }{4}$
  • D
    None of these
✓

Answer

Correct option: A.
Perpendicular
a
(a) Mid point of $(a\,\cos \alpha ,a\sin \alpha )$ and $(a\cos \beta ,a\sin \beta )$ is $P\,\left( {\frac{{a(\cos \alpha + \cos \beta )}}{2},\frac{{a(\sin \alpha + \sin \beta )}}{2}} \right)$

Slope of line $AB$ 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 $OP$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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