The magnetic field produced by an infinite wire with current $I$ is
$\begin{align*}\mathbf{B} = \frac{\mu_0 I}{2 \pi s} \hat{\mathbf{s}}\end{align*}$
where $\hat{\mathbf{s}}$ is a unit vector in the azimuthal direction determine by the right-hand rule. So, the magnetic field at $P$ due to the wire in the $\hat{\mathbf{x}}$ direction will be out of the page, and the magnetic field at $P$ due to the wire in the $\hat{\mathbf{y}}$ direction will be into of the page, and both will have the same magnitude, so they will cancel. Therefore, answer (E) is correct.

Solution to 2008 Problem 38