The lines of the form $\mathrm{xcos\varphi }+\mathrm{ysin\varphi }=\mathrm{p}$ are chords of the hyperbola $4{\mathrm{x}}^2 –{\mathrm{y}}^2 =4{\mathrm{a}}^2$which subtend a right angle at the centre of the hyperbola. If these chords touch a circle with centre at (0,0), then the radius of that circle is

$$\begin{gathered} \mathrm{Given} \mathrm{hyperbola} 4{\mathrm{x}}^2-{\mathrm{y}}^2=4{\mathrm{a}}^2 — ① \\ \mathrm{Given} \mathrm{chord} \mathrm{xcos}\cancel{\mathrm{\theta }}+\mathrm{y} \sin \cancel{\mathrm{\theta }}=\mathrm{P} — ② \\ \mathrm{Homogeniosing} ① \mathrm{with} \mathrm{help} \mathrm{of} ② \mathrm{then} \\ 4{\mathrm{x}}^2-{\mathrm{y}}^2=4{\mathrm{a}}^2{\left(\frac{\mathrm{xcos}\cancel{\mathrm{\theta }}+\mathrm{y} \sin \cancel{\mathrm{\theta }}}{\mathrm{P}}\right)}^2 \\ \Rightarrow 4{\mathrm{x}}^2-{\mathrm{y}}^2=\frac{4{\mathrm{a}}^2}{{\mathrm{P}}^2}({\mathrm{x}}^2 {\cos }^2\cancel{\mathrm{\theta }})+\frac{4{\mathrm{a}}^2}{{\mathrm{P}}^2}{\mathrm{y}}^2{\sin }^2\cancel{\mathrm{\theta }}+\frac{8{\mathrm{a}}^2}{{\mathrm{P}}^2}(\mathrm{xysin}\cancel{\mathrm{\theta } } \cos \cancel{\mathrm{\theta }}) \\ \mathrm{It} \mathrm{subtend} \mathrm{a} \mathrm{right} \mathrm{angle} \mathrm{at} (0, 0) \mathrm{then} \\ {\mathrm{x}}^2\mathrm{coeff} +{\mathrm{y}}^2 \mathrm{coeff}=0 \end{gathered}$$

$$\begin{gathered} \Rightarrow \left(\frac{4{\mathrm{a}}^2}{{\mathrm{P}}^2} {\cos }^2\cancel{\mathrm{\theta }}-4\right)+\left(\frac{4{\mathrm{a}}^2}{{\mathrm{P}}^2}{\sin }^2\cancel{\mathrm{\theta }}+1\right)=0 \\ \Rightarrow \frac{4{\mathrm{a}}^2}{{\mathrm{P}}^2}({\cos }^2\cancel{\mathrm{\theta }}+{\sin }^2\cancel{\mathrm{\theta }})=3 \\ \Rightarrow 3{\mathrm{P}}^2=4{\mathrm{a}}^2\left(1\right) \\ \Rightarrow \mathrm{P}=\frac{2\mathrm{a}}{\sqrt{3}} \\ \therefore ②\Rightarrow \mathrm{xcos}\cancel{\mathrm{\theta }}+\mathrm{ysin}\cancel{\mathrm{\theta }} \frac{2\mathrm{a}}{\sqrt{3}}=0 — ③ \\ \mathrm{If} \mathrm{it} \mathrm{is} \mathrm{a} \mathrm{tangent} \mathrm{to} \mathrm{circle} \mathrm{with} \mathrm{centre} (0, 0) \mathrm{then} \\ \mathrm{r}=\mathrm{d}=\perp \mathrm{be} \mathrm{distance} \mathrm{from} (0, 0) \mathrm{to} ③ \\ \Rightarrow \mathrm{r}=\frac{\left|{\displaystyle \frac{-2\mathrm{a}}{\sqrt{3}}}\right|}{\sqrt{{\cos }^2\cancel{\mathrm{\theta }}+{\sin }^2\cancel{\mathrm{\theta }}}} \Rightarrow \mathrm{r}=\frac{2\mathrm{a}}{\sqrt{3}} \end{gathered}$$