Question

If $l$ denotes the semi-latus rectum of the parabola $y^2=4ax$ and $SP$ and $SQ$ denote the segments of any focal chord $PQ$, $S$ being the focus, then $SP$, $l$ and $SQ$ are in the relation

• A. A.P
• B. G.P
• C. H.P
• D. $l^2={SP}^2+{SQ}^2$

Answer 1

The correct option is C H.P

Let $PQ$ be focal chord of the parabola $y^2=4ax$ with focus $S(a,0)$
$P(a{t}_1^2,2a{t}_1),Q(a{t}_2^2,2a{t}_2)$ are such that $SP$ and $SQ$ are segments of focal chord
$\Rightarrow t_1{t}_2=-1$
$SP=a+a{t}_1^2=a(1+t_1^2)$
$SQ=a(t_2^2+1)=a(\frac{1}{t_1^2}+1)=a\left(\frac{1+t_1^2}{t_1^2}\right)[\because t_1{t}_2=-1]$
$\therefore (\frac{1}{SP}+\frac{1}{SQ})=\frac{1}{a(t_1^2+1)}+\frac{t_1^2}{a(t_1^2+1)}=\frac{1}{a}\left[\frac{t_1^2+1}{t_1^2+1}\right]=\frac{1}{a}$
$\Rightarrow \frac{SQ+SP}{SQ.SQ}=\frac{1}{a}\Rightarrow \frac{2SP.SQ}{SQ+SP}=2a=l$ (semi-latus rectum$=2a$)
$\therefore$ Hence $SP,l,SQ$ are in HP