The sum of the distance of the any point on the ellripse `3x^(2) + 4y^(2) = 24` from its foxi is

Options

A. 4
B. `4sqrt(2)`
C. `8sqrt(2)`
D. `2sqrt(2)`

Answer

To solve the problem of finding the sum of the distances from any point on the ellipse \(3x^2 + 4y^2 = 24\) to its foci, we will follow these steps:

### Step 1: Rewrite the equation of the ellipse in standard form
We start with the given equation of the ellipse:

\[ 3x^2 + 4y^2 = 24 \]

To convert this into standard form, we divide the entire equation by 24:

\[ \frac{3x^2}{24} + \frac{4y^2}{24} = 1 \]

This simplifies to:

\[ \frac{x^2}{8} + \frac{y^2}{6} = 1 \]

### Step 2: Identify the values of \(a\) and \(b\)
In the standard form of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we can identify:

- \(a^2 = 8\) which gives \(a = \sqrt{8} = 2\sqrt{2}\)
- \(b^2 = 6\) which gives \(b = \sqrt{6}\)

### Step 3: Calculate the distance to the foci
The foci of an ellipse are located at a distance \(c\) from the center, where \(c\) is calculated using the formula:

\[ c = \sqrt{a^2 - b^2} \]

Substituting the values of \(a^2\) and \(b^2\):

\[ c = \sqrt{8 - 6} = \sqrt{2} \]

### Step 4: Determine the sum of distances from any point on the ellipse to the foci
The sum of the distances from any point on the ellipse to the two foci is given by the formula \(2a\):

\[ \text{Sum of distances} = 2a = 2 \times 2\sqrt{2} = 4\sqrt{2} \]

### Final Answer
Thus, the sum of the distances from any point on the ellipse \(3x^2 + 4y^2 = 24\) to its foci is:

\[ \boxed{4\sqrt{2}} \]