Let P(0, 6) and Q(0, – 4) be distinct points on parabola ‘C’ with focus O(0, 0). Let ‘L’ and ‘a’ denotes the length of latusrectum and length of intercept on x–axis by parabola C respectively. Then which of the following is/are TRUE ?

$$\begin{gathered} y=mx+c is directrix of parabola \\ S\left(0,0\right),P\left(0,6\right),Q\left(0,-4\right) \\ SP=PM\Rightarrow 6=\left|\frac{c-6}{\sqrt{1+m^2}}\right|\_\_\_\_\_\_ \left(1\right) \\ SQ=QM{}^{|}\Rightarrow 4=\left|\frac{c+4}{\sqrt{1+m^2}}\right|\_\_\_\_\_\_ \left(2\right) \\ From \left(1\right) and \left(2\right) \Rightarrow c=0\left(or\right)-24 \\ m=0\left(or\right)\pm 2\sqrt{6} \\ c=0andm=0\left(neglected\right) \\ Directrix is y=2\sqrt{6}x-24\left(or\right)y=-2\sqrt{6}x-24 \\ Length of latusrectum = 4a=2\left(2a\right)=2\left|\frac{24}{5}\right|=\frac{48}{5} \\ Equation of parabola is SP = PM\Rightarrow \sqrt{x^2+y^2}=\left|\frac{2\sqrt{6}x+y+24}{5}\right| \\ For x–intercept of parabola put y = 0 \\ \Rightarrow x^2-96\sqrt{x}-576=0 \\ a=\left|x_1-x_2\right|=240 \end{gathered}$$