Question

Let $PQ$ be a focal chord of parabola $y^2=x.$ If the coordinates of $P$ is $(4,-2),$ then the slope of the tangent at $Q$ is

• A. $8$
• B. $-4$
• C. $\frac{1}{8}$
• D. $4$

Answer 1

The correct option is D $4$

Given parabola is $y^2=x$
Any point on the parabola is $(\frac{t^2}{4},\frac{t}{2})$
Comparing with the coordinates of $P\left(t\right),$
$(\frac{t^2}{4},\frac{t}{2})=(4,-2)\Rightarrow t=-4$
As $PQ$ is focal chord, so coordinates of $Q(-\frac{1}{t})$ is $(\frac{1}{4t^2},-\frac{1}{2t})=(\frac{1}{64},\frac{1}{8})$

Now, equation of tangent at $Q$ is $T=0$
$\Rightarrow y\times \frac{1}{8}=\frac{1}{2}(x+\frac{1}{64})\Rightarrow y=4x+\frac{1}{16}$
Hence, the slope of the tangent at $Q$ is $4$.