The sum of focal distances of any point on the ellipse `9x^(2) + 16y^(2) = 144` is

Options

A. 32 nits
B. 18 units
C. 16 units
D. 8 units

Answer

To find the sum of the focal distances of any point on the ellipse given by the equation \(9x^2 + 16y^2 = 144\), we can follow these steps:

### Step 1: Rewrite the equation in standard form
The given equation is:
\[ 9x^2 + 16y^2 = 144 \]
To convert it into standard form, we divide each term by 144:
\[ \frac{9x^2}{144} + \frac{16y^2}{144} = 1 \]
This simplifies to:
\[ \frac{x^2}{16} + \frac{y^2}{9} = 1 \]

### Step 2: Identify the values of \(a\) and \(b\)
From the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we can identify:
- \(a^2 = 16\) which gives \(a = 4\)
- \(b^2 = 9\) which gives \(b = 3\)

### Step 3: Calculate the length of the major axis
The length of the major axis of an ellipse is given by \(2a\):
\[ \text{Length of major axis} = 2a = 2 \times 4 = 8 \]

### Step 4: Conclusion
The sum of the focal distances of any point on the ellipse is equal to the length of the major axis, which we have calculated to be:
\[ \text{Sum of focal distances} = 8 \]

Thus, the final answer is:
\[ \boxed{8} \]

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