Question

$PS{P}^{\prime }$ is a focal chord of the parabola $\frac{l}{r}=1-cos\theta$ and $SL$ is semi latus rectum, then $SP,SL,S{P}^{\prime }$ are in

• A. A.P.
• B. G.P
• C. H.P
• D. A.G.P

Answer 1

The correct option is C H.P

Let $P=(a{t}^2,2at)\therefore P^{\prime }=(\frac{a}{t^2},-\frac{2a}{t})$
$S=(a,0),L=(a,2a)$
Now $SP=\sqrt{(a-a{t}^2{)}^2+(2at{)}^2}=a+a{t}^2$
$S{P}^{\prime }=\sqrt{{(a-\frac{a}{t^2})}^2+{\left(\frac{-2a}{t}\right)}^2}=a+\frac{a}{t^2}=\frac{a+a{t}^2}{t^2}$
Now clearly $\frac{1}{SP}+\frac{1}{S{P}^{\prime }}=\frac{2}{SL}$
Hence $SP,SL,SP$ are in H.P
In this case only origin has been shifted. But the property of the parabola will remain same.