Question

If the point $P(4,-2)$ is one end of the focal chord $PQ$ of the parabola $y^2=x$, then the slope of the tangent at $Q$ is

Answer 1

For the given parabola $y^2=x,$
the value of $a=\frac{1}{4}$
Hence any point on the parabola will be $(\frac{t^2}{4},\frac{t}{2})$
Given $P\equiv (4,-2)$
Comparing it with parametric coordinates, we get
$t=-4$

Now, using the focal chord property,
Coordinates of $Q\equiv (\frac{1}{4t^2},-\frac{1}{2t})$
$\therefore Q\equiv (\frac{1}{64},\frac{1}{8})$
$\therefore$ Equation of tangent will be
$\frac{y}{8}=\frac{1}{2}(x+\frac{1}{64})$
Hence, required slope will be $4.$