The ordinates of the points P and Q on the parabola with focus ( 3,...

Given,

Focus $( 3, 0 )$, Directrix, $x = - 3$

So, the equation of parabola will be, ${y}^{2} = 12x$

Now taking the parametric point $\left(3{t}^{2}, 6t\right)$ and plotting the diagram we get,

Now we know that $R ( \alpha, \beta )$ will be the aithematic mean of ordinate and geometric mean of abscissa
will be,

Hence, $\beta = \dfrac{6\left({t}_{1} + {t}_{2}\right)}{2} and \alpha = \sqrt{3{{t}_{1}}^{2} \times 3{{t}_{2}}^{2}} = 3{t}_{1}{t}_{2}$

So, putting the value in $\dfrac{{\beta}^{2}}{\alpha}$, $⇒ \dfrac{{\beta}^{2}}{\alpha} = \dfrac{9{\left({t}_{1} + {t}_{2}\right)}^{2}}{3{t}_{1}{t}_{2}} = \dfrac{9{\left(4{t}_{2}\right)}^{2}}{3 \times 3{{t}_{2}}^{2}} = 16\left(∵ \dfrac{6{t}_{1}}{6{t}_{2}} = 3, given\right)$