If one end of a focal chord of the parabola, $${y^2} = 16x$$   is at $$\left( {1,\,4} \right),$$  then the length of this focal chord is :

Solution :
$$\because {y^2} = 16x\,\, \Rightarrow a = 4$$
One end of focal of the parabola is at (1, 4)
$$\because \,y$$ co-ordinate of focal chord is $$2\,at$$
$$\eqalign{ & \therefore 2\,at = 4 \cr & \Rightarrow t = \frac{1}{2} \cr} $$
Hence, the required length of focal chord
$$ = a{\left( {t + \frac{1}{t}} \right)^2} = 4 \times {\left( {2 + \frac{1}{2}} \right)^2} = 25$$