The product of eccentricities of two conics is unity, one of them can be a/an

Options

A. parabola
B. ellipse
C. hyperbola
D. circle

Answer

To solve the problem, we need to analyze the relationship between the eccentricities of two conics given that their product is unity.

### Step-by-Step Solution:

1. **Understanding Eccentricity**:
- The eccentricity (E) of a conic section helps us determine its type:
- For a circle, E = 0
- For an ellipse, 0 < E < 1
- For a parabola, E = 1
- For a hyperbola, E > 1

2. **Given Condition**:
- We are given that the product of the eccentricities of two conics is equal to 1:
\[ E \cdot E' = 1 \]
- Here, E and E' are the eccentricities of the two conics.

3. **Case Analysis**:
- **Case 1**: If both eccentricities are equal to 1, then:
\[ E = 1 \quad \text{and} \quad E' = 1 \]
- This means both conics are parabolas.

- **Case 2**: If one eccentricity is greater than 1 and the other is less than 1:
- Assume \( E > 1 \) and \( E' < 1 \).
- In this case, the conic with \( E > 1 \) must be a hyperbola, and the conic with \( E' < 1 \) must be an ellipse.
- Conversely, if \( E < 1 \) and \( E' > 1 \), the roles are reversed.

4. **Conclusion**:
- From the analysis, we can conclude:
- If the product of the eccentricities of two conics is unity, one of the conics can be a hyperbola and the other can be an ellipse.
- If both eccentricities are equal to 1, both conics are parabolas.

### Final Answer:
One of the conics can be a hyperbola, while the other can be an ellipse.