Question

The normal to the parabola $y^2=4x$ at $P\left(9,6\right)$ meets the parabola again at $Q.$ If the tangent at $Q$ meets the directrix at $R,$ then the slope of another tangent drawn from point $R$ to this parabola is

• A. $11$
• B. $\frac{11}{3}$
• C. $\frac{3}{11}$
• D. $3$

Answer 1

Answer: (B)

Parameter corresponding to $P$ is $t_1=3$
Hence, the parameter corresponding to $Q$ is $t_2=-t_1-\frac{2}{t_1}=-3-\frac{2}{3}=-\frac{11}{3}$
Let, $S\left(t_3\right)$ be the point where another tangent from $R$ touches the parabola
Now, if the tangent at $Q\left(t_2\right)$ and $S\left(t_3\right)$ meet on the directrix at $R$ then $t_2{t}_3=-1$
$\Rightarrow t_3=\frac{3}{11}$
So the equation of the tangent at $S$ is $\frac{3}{11}y=x+\frac{9}{121}$
Hence, the slope of the tangent at $S$ is $\frac{11}{3}$