The gravitational field in a region is given by $\vec{I}=(4 \hat{i}+\hat{j})$ N/kg. Work done by this field is zero when the particle is moved along the line

y + 4x = 2

Work done will be zero when displacement is perpendicular to the filed.

The field makes an angle

$\theta_1=\tan ^{-1}\left(\frac{1}{4}\right)$ with positive x-axis

While the line y + 4x = 2 makes an angle

$\theta_2=\tan ^{-1}(-4)$ with positive x-axis

$\theta_1+\theta_2=90^{\circ}$ i.e. the line y + 4x = 2 is perpendicular to $\vec{I}$