The potential in an electric field varies as $V = (x^2 - y^2)$. The electric lines of force in the $X-Y$ plane are

$(a)$ Consider an arbitrary electrostatic field configuration. $A$ small test charge is placed at a null point (i.e.,where $E = 0$) of the configuration. Show that the equilibrium of the test charge is necessarily unstable.
$(b)$ Verify this result for the simple configuration of two charges of the same magnitude and sign placed a certain distance apart.

Two point charges $+8q$ and $-2q$ are placed at $x = 0$ and $x = L$ respectively. The net electric field due to these two point charges will be zero at a point on the $x$-axis at:

Three equal charges $+q$ each are placed at the three vertices of an equilateral triangle. The electric field at the centroid of the triangle is . . . . . . . ('$r$' is the length of the side of the triangle).

The figures below show regular hexagons with charges at the vertices. In which of the following cases is the electric field at the center $NOT$ zero?

The electric field at the centre $O$ of a semicircle of radius $a$ having a linear charge density $\lambda$ is given by: