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

Options

A. Length of latusrectum
B. Length of major-axis
C. Length of minor-axis
D. Length of semi-latusretum

Explanation

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

### Step 1: Understand the Definition of Focal Radii
The focal radii of an ellipse are the distances from a point on the ellipse to each of its foci. For an ellipse with foci at points F and F', if P is a point on the ellipse, then the focal radii are PF and PF'.

**Hint:** Recall that the foci of an ellipse are located along the major axis, symmetrically about the center.

### Step 2: Set Up the Ellipse
Consider an ellipse centered at the origin (0,0) with the major axis along the x-axis. The standard form of the equation of the ellipse is:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where \( a \) is the semi-major axis and \( b \) is the semi-minor axis.

**Hint:** Identify the values of \( a \) and \( b \) based on the ellipse's dimensions.

### Step 3: Identify the Foci
The foci of the ellipse are located at points \( (c, 0) \) and \( (-c, 0) \), where \( c = \sqrt{a^2 - b^2} \).

**Hint:** Remember the relationship between \( a \), \( b \), and \( c \) in an ellipse.

### Step 4: Calculate the Focal Radii
For any point \( P(x, y) \) on the ellipse, the distances to the foci are:
- Distance to focus F: \( PF = \sqrt{(x - c)^2 + y^2} \)
- Distance to focus F': \( PF' = \sqrt{(x + c)^2 + y^2} \)

**Hint:** Use the distance formula to express the focal radii in terms of coordinates.

### Step 5: Sum of Focal Radii
The sum of the focal radii is:
\[ PF + PF' = \sqrt{(x - c)^2 + y^2} + \sqrt{(x + c)^2 + y^2} \]
However, by the definition of an ellipse, this sum is constant and equals the length of the major axis, which is \( 2a \).

**Hint:** Recall the property of ellipses that states the sum of the distances from any point on the ellipse to the foci is constant.

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

**Final Answer:** The sum of the focal radii of any point on an ellipse is equal to \( 2a \).