Question

PQ is the chord of contact of tangents from T to a parabola. If PQ be normal at P, then the directrix of the parabola divides PT in the ratio

• A. 1: 2
• B. 2: 1
• C. 1 : 1
• D. None of these

Answer 1

The correct option is C 1 : 1

Let Parabola: $y^2=4ax...............\left(1\right)$

$PQ$ is chprd of contact from $T.$

And also $PQ$ is normal to $\left(1\right)$ at $P$

Then, let $P$ be $(a{t}^2,2at),$

$=>Q:\left(a\right(-t-\frac{2}{t}{)}^2,2a(-t-\frac{2}{t})$

Now, equation of normal at $P\left(t\right)$

$y-tx=2at+a{t}^3......................\left(2\right)$

Chord of contact, and $T(h,k)$ (say)

$=>yk=2a(x+h)$

$=>y-\frac{2a}{k}x=\frac{2ah}{k}................\left(3\right)$

$\left(2\right)$ and $\left(3\right)$ represent the same chord, $=>\frac{-2a}{k}=t$

$=>k=\frac{-2a}{t}$ and $\frac{2ah}{k}=2at+{at}^3$

$=>\frac{2ah}{\frac{-2a}{t}}=2at+{at}^3$

So, $T(-2a-a{t}^2,\frac{-2a}{t})$ $=>h=-2a-a{t}^2$

Line passing through $P$ and $T=>(y-2at)=\frac{(\frac{-2a}{t}-2at)}{(-2a-{at}^2-{at}^2)}(x-{at}^2)$

$=>ty=2a{t}^2+x-a{t}^3...............\left(4\right)$

intersection of $x=-a$ and $\left(4\right),$ $A(-a,\frac{2at-a-{at}^3}{t})$

$=>-a=\frac{m({at)}^2+n(-2at-{at}^3)}{m+n}$ and $\frac{2at-a-{at}^3}{t}=\frac{m\left(2at\right)+n\left(\frac{-2a}{t}\right)}{m+n}$

$=>m+n=2$ and $m=1,n=1.$