An element `A` decays into element `C` by a two-step process :
`A rarr B + ._2 He^4`
`B rarr C + 2 e overline`
Then.

Options

A. A and C are isotopes
B. A and C are isobars
C. B and C are isotopes
D. A and B are isobars

Explanation

To solve the problem, we need to analyze the decay process of element A into element C through an intermediate element B. The decay process is given as follows:

1. **Decay of A to B**:
\[ A \rightarrow B + \alpha \text{ (where } \alpha \text{ is } ^4_2He) \]
Here, element A decays into element B and emits an alpha particle (which has a mass number of 4 and atomic number of 2).

2. **Decay of B to C**:
\[ B \rightarrow C + 2\beta^- \]
In this step, element B decays into element C and emits two beta particles (electrons).

### Step 1: Determine the Mass and Atomic Numbers
- Let the mass number of element A be \( A \) and its atomic number be \( Z \).
- After the first decay:
- Mass number of B: \( A - 4 \)
- Atomic number of B: \( Z - 2 \)

### Step 2: Analyze the Decay of B
- For the second decay:
- Mass number of C: Since B decays to C and emits 2 beta particles (which do not change the mass number), the mass number of C remains the same as B.
- Therefore, mass number of C: \( A - 4 \)
- Atomic number of C: Since B has an atomic number of \( Z - 2 \) and emits 2 beta particles (which increase the atomic number by 2), the atomic number of C will be:
\[ Z - 2 + 2 = Z \]

### Step 3: Summarize the Relationships
- We have:
- Mass number of B: \( A - 4 \), Atomic number of B: \( Z - 2 \)
- Mass number of C: \( A - 4 \), Atomic number of C: \( Z \)

### Step 4: Identify Relationships Between Elements
- Since the atomic number of C is \( Z \) and that of A is also \( Z \), elements A and C have the same atomic number, making them isotopes.
- Since B has a different atomic number than A and C, it cannot be an isotope of either.
- B and C have the same mass number (\( A - 4 \)), making them isobars.

### Conclusion
- A and C are isotopes (same atomic number, different mass numbers).
- B and C are isobars (same mass number, different atomic numbers).
- A and B are neither isotopes nor isobars.

### Final Answer
The correct option is that A and C are isotopes.