Question

two mutually perpendicular long conducting wire carrying currents I1 and I2 lie in one plane. Locus of point at which magnetic induction is zero

Answer 1

I_1|x| = I_2|y|

Answer 2

Assume wires are along x and y axes. Magnetic field from wire 1 (along x-axis) at P(x,y) is $B_1 = \frac{\mu_0 I_1}{2\pi |y|}$. Magnetic field from wire 2 (along y-axis) at P(x,y) is $B_2 = \frac{\mu_0 I_2}{2\pi |x|}$. For zero net magnetic field, $B_1$ and $B_2$ must be equal in magnitude and opposite in direction. Equating magnitudes gives $\frac{\mu_0 I_1}{2\pi |y|} = \frac{\mu_0 I_2}{2\pi |x|}$, which simplifies to $I_1|x| = I_2|y|$. The directions of fields cancel in two opposite quadrants, leading to two lines passing through the origin, $y = \pm \frac{I_1}{I_2} x$.