Two long mutually perpendicular conductors carrying currents $I_1$ and $I_2$ lie in one plane. Find the locus point at which the magnetic induction is zero. 

In first quadrant magnetic field due to $I_1$ is outwards and due to $I_2$ is inwards. So ,net magnetic field due to $I_1$ is inwards and due to $I_2$ magnetic field is outwards. hence, only in first and third quadrants filed may be zero. 

Let magnetic field is zero at point $P\left(xy\right)$, then

$$\begin{gathered} B_1=B_2 \\ \therefore \frac{{\mu }_0}{2\mathrm{\pi }}\frac{I_1}{y}=\frac{{\mu }_0}{2\mathrm{\pi }}\frac{I_2}{x} \\ \therefore y=\frac{I_1}{I_2}x \end{gathered}$$