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Electric charges are fundamental properties of matter, and they come in two types: positive and negative. Like charges repel each other, while opposite charges attract. The charge of an electron is negative and is often noted as \(-e\), while a proton carries a positive charge of \(+e\). Several basic principles govern electric charges:

• Quantization: Charge is quantized, meaning it exists in discrete amounts. The smallest unit of charge is that of an electron or proton, referred to as elementary charge.
• Conservation: In a closed system, the total electric charge remains constant over time. This means the algebraic sum of all charges remains unchanged.

In the exercise given, the two charges involved are \(q\) and \(-2q\). These charges are situated a fixed distance apart, and the principles of electric potential apply to them extensively.

The locus of zero potential refers to the set of all points in a field where the electric potential is zero. When dealing with multiple charges, each charge contributes to the potential at a point. The condition for zero potential depends on the spatial arrangement and magnitudes of these charges.In our exercise, we have a charge \(q\) at the origin and a charge \(-2q\) at some distance away. The locus of points with zero potential will depend on their distances from these charges. By equating the potentials from the charges at a point to zero, we find the equation of the locus which satisfies the zero potential condition. Through algebraic manipulation, this locus is mathematically expressed, leading us to the correct option (c) of the exercise.

The electric potential \(V\) at a point due to a point charge \(Q\) is given by \(V = \frac{kQ}{r}\), where \(k\) is Coulomb's constant and \(r\) is the distance from the charge to the point. This relationship highlights the inverse nature of potential and distance: as the distance increases, the potential decreases.For our two charges, this relationship forms the basis for establishing the zero potential condition: \(q\)'s potential is balanced by \(-2q\)'s potential at some point so that the net potential is zero. Using geometric distance formulas, we identify how the potentials interplay due to their differing distances. Solving through the condition \(\frac{kq}{r_1} = \frac{2kq}{r_2}\) leads to this locus equation. Understanding the inverse distance-potential relationship is key to interpreting the locus of zero potential.

Quadratic equations frequently appear in physics, often in contexts that involve geometric or dynamic systems. They are expressed in the form \(ax^2 + bx + c = 0\) and can describe paths, forces, or potentials in systems.In this case, simplifying the derived locations of zero potential using coordinate systems translates into a quadratic form: \(x^2 + y^2 + 4ax - 12a^2 = 0\). This equation describes a conic section, indicating the path or shape that would satisfy the zero potential condition between the charges. Quadratic equations thus capture complex relationships compactly, linking geometrical loci to physical scenarios.