Question

If a straight line through origin bisects the line passing through the given points $A(a\cos \alpha ,a\sin \alpha )$ and $B(a\cos \beta ,a\sin \beta )$, then the lines

• A. are perpendicular
• B. are parallel
• C. have an angle between them of $\frac{\pi }{4}$
• D. None of these

Answer 1

Answer: (A)

are perpendicular

$P$ bisects $AB$ thus $P$ is midpoint of $AB$
$P\equiv (\frac{a\cos \alpha +a\cos \beta }{2},\frac{a\sin \alpha +a\sin \beta }{2})$
Line $AB$ has slope$m_1=\frac{a\sin \beta -a\sin \alpha }{a\cos \beta -a\cos \alpha }$
line $OP$ has slope $m_2=\frac{\frac{a}{2}\times (\sin \beta +\sin \alpha )-0}{\frac{a}{2}\times (\cos \beta +\cos \alpha )-0}$
$m_1\times m_2=\frac{{\sin }^2\beta -{\sin }^2\alpha }{{\cos }^2\beta -{\cos }^2\alpha }=-1$
therefore $AB$ and $OP$ are perpendicular to each other.