Introduction:
The problem describes the motion of a particle under the influence of a central force in a plane polar coordinate system. The position of the particle is given by the equation r = a e^(kθ), where a and k are constants. The angular momentum of the particle is denoted by L, and the total energy of the system is zero. We are required to find the potential energy V(r) in terms of m, L, and k.

Understanding the problem:
To solve the problem, we need to analyze the given information and understand the concepts involved:

1. Central force: A central force acts on a particle toward or away from a fixed point, which is the origin in this case. The force is always directed along the radial direction.

2. Plane polar coordinates: In a plane polar coordinate system, the position of a particle is described by its radial distance (r) from the origin and the angle (θ) it makes with a reference axis.

3. Angular momentum (L): Angular momentum is a vector quantity defined as the cross product of the position vector (r) and the linear momentum vector (p). In this case, the angular momentum (L) is a constant.

4. Total energy (0): The total energy of the system is the sum of kinetic energy (T) and potential energy (V). Since the total energy is zero, this implies that the kinetic energy and potential energy have equal magnitudes but opposite signs.

Solution:
To find the potential energy V(r), we can use the concept of conservation of energy and the expression for angular momentum.

1. Conservation of energy:
- The total energy of the system is given by E = T + V.
- Since the total energy is zero (E = 0), we have T = -V.

2. Expression for kinetic energy:
- The kinetic energy of the particle can be expressed as T = (1/2) m (dr/dt)^2.
- Since the motion is in a plane, the velocity components can be written as v_r = dr/dt and v_θ = r(dθ/dt).
- The radial velocity (v_r) can be obtained by differentiating the given equation for r with respect to time.

3. Expression for potential energy:
- As per the conservation of energy, V = -T.
- Substituting the expression for T and simplifying, we can obtain V as a function of r, m, L, and k.

Conclusion:
By applying the principles of conservation of energy and understanding the concepts of central force motion in a plane polar coordinate system, we can derive the potential energy V(r) in terms of m, L, and k. This solution helps us analyze the motion and behavior of the particle under the given conditions.