Two mutually perpendicular conductors carrying currents $I_1$ and $I_2$ lie in one plane. Locus of the point at which the magnetic induction is zero, is a
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Magnetic field in $X-Y$ plane at position $(x, y)$ due to both wires must be zero current carrying wire with current i 1 is taken along $X$ axis now the magnetic field at the given point is $B_1=\frac{\mu_0 i_1}{2 \pi y}$
now at the same point the magnetic field due to other wire placed along $Y$ axis is given by
$B_2=\frac{\mu_0 i_2}{2 \pi x}$
now we have to make the magnitude of magnetic field due to both wires same
$B_1=B_2$
$\frac{\mu_0 i_1}{2 \pi y}=\frac{\mu_0 i_2}{2 \pi x}$
now by solving above we have
$y=\frac{i_1}{i_2} x$
so its a straight line with slope $i1/i2$ and passing from origin
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