What is the sum of focal radii of any point on an ellise equal to?

Options

A. Length of latus rectum
B. Length of major axis
C. Length of minor axis
D. Length of semi latus rectum

Answer

To find the sum of the focal radii of any point on an ellipse, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Ellipse**:
An ellipse is defined as the set of all points such that the sum of the distances from two fixed points (the foci) is constant.

2. **Identifying the Axes**:
- The major axis is the longest diameter of the ellipse, and the minor axis is the shortest.
- The center of the ellipse is at the origin (0,0).

3. **Locating the Foci**:
- Let the foci of the ellipse be denoted as F and F'.
- The distance from the center to each focus is denoted as 'c'.

4. **Understanding Focal Radii**:
- For any point P on the ellipse, the distances from P to each focus (F and F') are called the focal radii.
- Let the distance from point P to focus F be denoted as PF and the distance to focus F' be denoted as PF'.

5. **Applying the Definition of an Ellipse**:
- By the definition of an ellipse, the sum of the distances from any point P on the ellipse to the two foci is constant.
- This constant is equal to the length of the major axis of the ellipse.

6. **Conclusion**:
- Therefore, the sum of the focal radii (PF + PF') for any point P on the ellipse is equal to the length of the major axis, which is represented as 2a (where 'a' is the semi-major axis).

### Final Answer:
The sum of the focal radii of any point on an ellipse is equal to the length of the major axis, which is \( 2a \).

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