1. Work in the non-inertial frame of the wire

Since the wire is accelerated horizontally with acceleration $a$, in the frame of the wire the bead experiences a pseudo force of magnitude $ma$ opposite to the acceleration.

So the forces on the bead are:

• Gravity: $mg$ downward
• Pseudo force: $ma$ horizontally opposite to motion of wire
• Normal reaction: perpendicular to the wire

For the bead to remain at rest relative to the wire, the resultant of $mg$ and pseudo force must be normal to the curve, i.e. it should have no component along the tangent.

2. Equation of the parabola

The wire is shaped as $y = kx^2$

At any point, slope of the tangent is $\frac{dy}{dx} = 2kx$

So tangent makes an angle such that its slope is $2kx$.

3. Condition for equilibrium along the tangent

Let the wire accelerate toward positive $x$. Then pseudo force acts toward negative $x$.

Effective acceleration in the wire frame is:

• horizontal: $a$ toward $-x$
• vertical: $g$ downward

Thus the effective gravity vector has components $(-a,-g)$

For equilibrium, this effective gravity must be perpendicular to the tangent.

A tangent vector is proportional to $(1,2kx)$

Dot product must be zero: $(-a,-g)\cdot(1,2kx)=0$

$-a-2gkx=0$

$2gkx=-a$

So the coordinate is $x=-\frac{a}{2gk}$

The question asks for the distance from the y-axis, so we take magnitude: $|x|=\frac{a}{2gk}$

4. Final answer

The new equilibrium position is at a distance $\boxed{\frac{a}{2gk}}$ from the $y$-axis.

So the correct option is B.