*PSQ* is a focal chord of the parabola *y*^{2} = 8*x*. If *SP* = 6, then write *SQ*.

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#### Solution

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 *y*^{2} = 8*x** *with

\[y^2 = 4ax\]

*a*= 2

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

Given:

*SP*= 6\[\therefore \sqrt{\left( 2 - 2 t^2 \right)^2 + \left( 4t \right)^2} = 6\]

\[ \Rightarrow t^4 + 2 t^2 - 8 = 0\]

\[ \Rightarrow t^2 = 2\]

\[ \Rightarrow t^4 + 2 t^2 - 8 = 0\]

\[ \Rightarrow t^2 = 2\]

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

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

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