Question

PSQ is a focal chord of the parabola y² = 8x. If SP = 6, then write SQ.

Answer 1

The coordinates of the focal chord are $P\left(a{t}^2,2at\right)\mathrm{and}Q\left(\frac{a}{t^2},\frac{-2a}{t}\right)$.
Comparing y² = 8x with $y^2=4ax$:
a = 2

Therefore, the coordinates of the focus S is $\left(2,0\right)$.

Given:
SP = 6

$$\begin{gathered} \therefore \sqrt{{\left(2-2t^2\right)}^2+{\left(4t\right)}^2}=6 \\ \Rightarrow t^4+2t^2-8=0 \\ \Rightarrow t^2=2 \end{gathered}$$

Thus, we have:

SQ = $\sqrt{{\left(2-\frac{2}{t^2}\right)}^2+\left(\frac{4}{t^2}\right)}$=$\sqrt{{\left(2-\frac{2}{2}\right)}^2+\left(\frac{4}{2}\right)}$ = 3