Question

In the parabola $y^2=4ax$, the tangent at the point $P$, whose abscissa is equal to the length of latus rectum meets the axis in $T$ & the normal at $P$ cuts the parabola again in $Q$. Then if the ratio of $PT:PQ=3:5$, write 1 otherwise write 0 ?

Answer 1

Let point on parabola $y^2=4ax$ is $P(a{t}^2,2t)$
Given $a{t}^2=4a\Rightarrow t=\pm 2$
Taking positive $t$, $t=2$
$P(4a,4a)$
Equation of tangent at $P$ is $2y=x+4a$
If it intersects x-axis at $T$ then $T(-4a,0)$
Normal at $(4a,4a)$ meets again parabola at
$Q(a{t}_2^2,2a{t}_2)$ (using $t_2=-t_1-\frac{2}{t_1}=-3)$
$\therefore Q(9a,-6a)$
Now, $P(4a,4a),T(-4a,0),Q(9a,6a)$
$PT=\sqrt{(4a+4a{)}^2+(4a{)}^2}=\sqrt{80a^2}$
$PQ=\sqrt{(4a+9a{)}^2+(4a+6a{)}^2}=\sqrt{125a^2}$
$\therefore \frac{PT}{PQ}=\sqrt{\frac{80a^2}{125a^2}}=\frac{4}{5}$