Three infinitely long wires with linear charge density λ are placed...

Potential due to an infinite wire is $V=2 k \lambda \ln r$, where $r$ is the distance from the wire. Taking the point in space $P(x, y, z)$ Distance from wire along $x$-axis is $r_x=\sqrt{y^2+z^2}$ Distance from wire along $y$-axis is $r_y=\sqrt{x^2+z^2}$ Distance from wire along $z$-axis is $r_z=\sqrt{x^2+y^2} \Rightarrow$ Potential at $P$ due to wire along $x$-axis is $V_x=2 k \lambda \ln r_x$ Potential at $P$ due to wire along $y$-axis is $V_y=2 k \lambda \ln r_y$ Potential at $P$ due to wire along $z$-axis is $V_z=2 k \lambda \ln r_z \Rightarrow$ Not potential at P=V=V_x+V_y+V_z or $\quad V=2 k \lambda \ln r_x+2 k \lambda \ln r_y+2 k \lambda \ln r_z$ i.e. $V=2 k \lambda \ln \left(r_x r_y r_z\right) or$$\begin{aligned} & V=2 k \lambda\ln\left(\sqrt{y^2+z^2} \sqrt{z^2+x^2} \sqrt{x^2+y^2}\right) \\ & =k \lambda \ln\left(y^2+z^2\right)\left(z^2+x^2\right)\left(x^2+y^2\right) \end{aligned}$$\Rightarrow$ For equipotential surface $\left(x^2+y^2\right)\left(y^2+z^2\right)\left(z^2+x^2\right)=\text{ constant }