We know that distance of a point from a line is:
$$
\begin{aligned}
& \mathrm{d}=\frac{\left|\mathrm{Ax}_1+\mathrm{By}_1+\mathrm{C}\right|}{\sqrt{\mathrm{A}_2+\mathrm{B}_2}} \\
\Rightarrow \quad \mathrm{p}=\frac{|0+0-1|}{\sqrt{\frac{1}{\mathrm{a}^2}+\frac{1}{\mathrm{~b}^2}}} \\
\Rightarrow \quad \frac{1}{\mathrm{a}^2}+\frac{1}{\mathrm{~b}^2}=\frac{1}{\left(\mathrm{p}^2\right)} \\
\Rightarrow \quad \frac{1}{\left(\mathrm{p}^2\right)}=\frac{\mathrm{b}^2+\mathrm{a}^2}{\mathrm{a}^2 \mathrm{~b}^2} \Rightarrow \mathrm{p}^2=\frac{\mathrm{a}^2 \mathrm{~b}^2}{\mathrm{a}^2+\mathrm{b}^2} \\
&
\end{aligned}
$$
As $\mathrm{a}^2, \mathrm{p}^2, \mathrm{~b}^2$ are in A.P.
$$
\Rightarrow \quad \mathrm{p}^2=\frac{\mathrm{a}^2+\mathrm{b}^2}{2}
$$
from (i) and (ii), we get
$$
\frac{a^2+b^2}{2}=\frac{a^2 b^2}{a^2+b^2}
$$
$\begin{array}{ll}\Rightarrow & \left(a^2+b^2\right)^2=2 a^2 b^2 \\ \Rightarrow & a^4+b^4+2 a^2 b^2=2 a^2 b^2 \\ \Rightarrow & a^4+b^4=0\end{array}$