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 \[P \left( a t^2 , 2at \right) a\text{ and } Q \left( \frac{a}{t^2}, \frac{- 2a}{t} \right)\] 

Comparing y² = 8x with

\[y^2 = 4ax\]

a = 2 

Therefore, the coordinates of the focus S is \[\left( 2, 0 \right)\] 

Given:
SP = 6 

\[\therefore \sqrt{\left( 2 - 2 t^2 \right)^2 + \left( 4t \right)^2} = 6\]
\[ \Rightarrow t^4 + 2 t^2 - 8 = 0\]
\[ \Rightarrow t^2 = 2\] 

Thus, we have:
SQ = \[\sqrt{\left( 2 - \frac{2}{t^2} \right)^2 + \left( \frac{4}{t^2} \right)}\] 

\[\sqrt{\left( 2 - \frac{2}{2} \right)^2 + \left( \frac{4}{2} \right)}\]