The difference of the focal distance of any point on the hyperbola `9x ^(2) -16 y ^(2) =144,` is

Options

A. 8
B. 7
C. 6
D. 4

Answer

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

### Step 1: Rewrite the Hyperbola Equation
First, we need to rewrite the hyperbola equation in standard form. The given equation is:
\[ 9x^2 - 16y^2 = 144 \]
To convert it to standard form, divide the entire equation 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 Parameters
In the standard form of the hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), we can identify:
- \(a^2 = 16\) which gives \(a = \sqrt{16} = 4\)
- \(b^2 = 9\) which gives \(b = \sqrt{9} = 3\)

### Step 3: Determine the Length of the Transverse Axis
The length of the transverse axis for a hyperbola is given by \(2a\). Therefore:
\[ \text{Length of Transverse Axis} = 2a = 2 \times 4 = 8 \]

### Step 4: Conclusion
The difference of the focal distances (the distance from any point on the hyperbola to each focus) is equal to the length of the transverse axis. Thus, the difference of the focal distances is:
\[ \text{Difference of Focal Distances} = 8 \]

### Final Answer
The difference of the focal distance of any point on the hyperbola \(9x^2 - 16y^2 = 144\) is \(8\).

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