Equation of $\text{L}$ is $\frac{\text{x}}{\text{a}}+\frac{\text{y}}{\text{b}}=1$ and its distance from the origin is $\frac{1}{\sqrt{\frac{1}{a^2}+\frac{1}{b^2}}}$

Let the axis be rotated through an angle $\theta$ and let $\left(\text{X,}\text{Y}\right)$ be the new coordinates of any point $\text{P}\left(\text{x,}\text{y}\right)$ in the plane, then new equation of $\text{L}$ will be

$\frac{\text{X}}{\text{p}}+\frac{\text{Y}}{\text{q}}=1$ and its distance from the origin is $\frac{1}{\sqrt{\frac{1}{p^2}+\frac{1}{q^2}}}$

As the origin is fixed, distance from origin will be same in both cases

$\frac{1}{{\text{a}}^2}+\frac{1}{{\text{b}}^2}=\frac{1}{{\text{p}}^2}+\frac{1}{{\text{q}}^2}$