Let L₁ : x + y = 0 and L₂ : x – y = 0 are tangent to a parabola whose focus is S(1, 2). If the length of latus rectum of the parabola can be expressed as $\frac{\mathrm{m}}{\sqrt{\mathrm{n}}}$ (where m and n are coprime them value of $\frac{\mathrm{m}+\mathrm{n}}{11}$ is

x+y=0,x-y=0 are two perpendicular tangents intersecting at (0,0)

The image of focusS(1,2) with respect to the tangent y=x is(2,1)

Equation of directrix is x-2y=0

Distance from S(1, 2) from the  directrix  is $$\begin{gathered} \frac{1}{2}\times \text{ L.R. }=\frac{3}{\sqrt{5}}=\Rightarrow \text{ Length of .Latus.Rectum. }=\frac{6}{\sqrt{5}} \\ m=6,n=5 \end{gathered}$$