Question

The lengths of the perpendicular from the points $(m^2,2m)$, $(m{m}^{\prime },m+m^{\prime })$ and $(m^{\prime 2},2m^{\prime })$ to the line $x+y+1=0$ form
$\cos \alpha +y\sin \alpha +\sin \alpha \tan \alpha =0$ are in

• A. an A.P.
• B. a G.P.
• C. a H.P.
• D. none of these

Answer 1

The correct option is B a G.P.

$p_1=\frac{m^2+2m+1}{\sqrt{2}},p_2=\frac{m{m}^{\prime }+m+m^{\prime }+1}{\sqrt{2}},p_3=\frac{m^{\prime 2}+2m^{\prime }+1}{\sqrt{2}}$
Then
$p_1{p}_3=p_2^2\Rightarrow p_1,p_2,p_3$ are in $G.P$