Question

If $PS{P}^{\prime }$ is a focal chord of the parabola $y^2=4ax$ and $SL$ is its semi latusrectum then $SP,SL,S{P}^{\prime }$ are in

• A. AP
• B. HP
• C. GP
• D. AGP

Answer 1

Answer: (B)

HP

Let $P=(a{t}^2,2at),\therefore P^{\prime }=(\frac{a}{t^2},-\frac{2a}{t})$
$S=(a,0),L=(a,2a)$
$SL=2a$
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, $SP,SL,S{P}^{\prime }$ are in H.P