The relationship between, the semi-major axis, seimi-minor axis and the distance of the focus from the centre of the ellipse is

Options

A. `a^(2) = b^(2) -c^(2)`
B. `a^(2) = b^(2) + c^(2)`
C. `b^(2) = a^(2) + c^(2)`
D. `c^(2) = a^(2) + b^(2)`

Answer

To find the relationship between the semi-major axis (a), semi-minor axis (b), and the distance of the focus (c) from the center of the ellipse, we can use the following steps:

### Step-by-Step Solution:

1. **Understand the Definitions**:
- The semi-major axis (a) is half the length of the major axis of the ellipse.
- The semi-minor axis (b) is half the length of the minor axis of the ellipse.
- The distance of the foci from the center of the ellipse is denoted as c.

2. **Use the Relationship Formula**:
- The relationship between these three quantities is given by the equation:
\[ c^2 = a^2 - b^2 \]
- Here, c is the distance from the center to each focus, a is the semi-major axis, and b is the semi-minor axis.

3. **Rearranging the Equation**:
- If we want to express a in terms of b and c, we can rearrange the equation:
\[ a^2 = b^2 + c^2 \]
- This shows that the square of the semi-major axis is equal to the sum of the square of the semi-minor axis and the square of the distance from the center to the focus.

4. **Conclusion**:
- Thus, the relationship between the semi-major axis, semi-minor axis, and the distance of the focus from the center of the ellipse is summarized by the equations:
\[ c^2 = a^2 - b^2 \quad \text{or} \quad a^2 = b^2 + c^2 \]