Question

If the tangents at $P$ and $Q$ on the parabola $y^2=4ax$ meets at $T$ and if $S$ is the focus of the parabola then, $SP,ST,SQ$ are in

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

Answer 1

The correct option is B G.P.

Let $P(a{t}_1^2,2a{t}_1)$, $Q(a{t}_2^2,2a{t}_2)$ then $T$ is $(a{t}_1{t}_2,a(t_1+t_2\left)\right)$ and $S(a,0)$
Let $a>0$,
Now focal distances
$SP=x_1+a=a(t_1^2+1)SQ=x_2+a=a(t_2^2+1)ST=\sqrt{(a-a{t}_1{t}_2{)}^2+a^2(t_1+t_2{)}^2}\Rightarrow S{T}^2=a^2(t_1^2+t_1^2{t}_2^2+t_2^2+1)\Rightarrow S{T}^2=a(t_1^2+1)\times a(t_2^2+1)\Rightarrow S{T}^2=SP\cdot SQ$

$\therefore SP,ST,SQ$ are in G.P