If one end of a focal chord of the parabola ${\mathrm{y}}^2=\frac{8}{a}\mathrm{x},(\mathrm{a}>0)$ is at (1, 4), then the length of this focal chord is

${\text{y}}^{\text{2}}\text{=}\frac{\text{8}}{\text{a}}\text{x\_\_\_\_\_\_}\left(\text{1}\right)$

(1) passes through (1, 4) $\Rightarrow \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}a\hspace{0.17em}=\hspace{0.17em}}\frac{\text{1}}{\text{2}}$

$\therefore {\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}y}}^{\text{2}}\text{=\hspace{0.17em}16x}$

$P(1,4)=\left(a{t}^2,2at\right)$

2at = 4

$\text{8t=4\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\Rightarrow \text{\hspace{0.17em}\hspace{0.17em}t\hspace{0.17em}=\hspace{0.17em}}\frac{\text{1}}{\text{2}}$

Length of focal chord $\text{PQ\hspace{0.17em}=\hspace{0.17em}a}{\left(\text{t\hspace{0.17em}+\hspace{0.17em}}\frac{\text{1}}{\text{t}}\right)}^{\text{2}}$

$\text{=\hspace{0.17em}4}{\left(\frac{\text{1}}{\text{2}}\text{+\hspace{0.17em}\hspace{0.17em}2}\right)}^{\text{2}}$

= 25