Question

If $x\cos \alpha +y\sin \alpha =p$ where, $p=\frac{{\sin }^2\alpha }{\cos \alpha }$ be a straight line, prove that perpendiculars $p_1,p_2$ and $p_3$ on this line from the points $(m^2,2m),(mm,m+m)$ and $(m^2,2m)$ respectively are in geometrical progression.

Answer 1

$p_1=\frac{m^2\cos \alpha +2m\sin \alpha (-{\sin }^2\alpha /\cos \alpha )}{\surd ({\cos }^2\alpha +{\sin }^2\alpha )}$
$p_1={\left(\frac{m\cos \alpha +\sin \alpha }{\surd \left(\cos \alpha \right)}\right)}^2$
Similarly $p_3={\left(\frac{m^{\prime }\cos \alpha +\sin \alpha }{\surd \left(\cos \alpha \right)}\right)}^2$
Also
$p_2=\frac{m{m}^{\prime }\cos \alpha +(m+m^{\prime })\sin \alpha -(-{\sin }^2\alpha /\cos \alpha )}{\surd ({\cos }^2\alpha +{\sin }^2\alpha )}$
or $p_2=\frac{1}{\cos \alpha }[m{m}^{\prime }{\cos }^2\alpha +(m+m^{\prime })\sin \alpha \cos \alpha +{\sin }^2\alpha ]$
or $p_2=\frac{(m\cos \alpha +\sin \alpha )}{\surd \left(\cos \alpha \right)}.\frac{m^{\prime }\cos \alpha +\sin \alpha }{\surd \left(\cos \alpha \right)}$
or $p_2=\sqrt{p_1.p_3}$ or $p_2^2=p_1{p}_3$
Hence $p_1,p_2,p_3$ are in $G.P.$