In the hyperbola`16 x^(2) - 9y^(2) =144`, the value of eccentricity is

Options

A. `(1)/(3)`
B. `(4)/(3)`
C. `(5)/(3)`
D. `(7)/(3)`

Answer

To find the eccentricity of the hyperbola given by the equation \( 16x^2 - 9y^2 = 144 \), we can follow these steps:

### Step 1: Rewrite the equation in standard form
We start with the equation of the hyperbola:
\[ 16x^2 - 9y^2 = 144 \]
To rewrite it in standard form, we divide the entire equation by 144:
\[ \frac{16x^2}{144} - \frac{9y^2}{144} = 1 \]
This simplifies to:
\[ \frac{x^2}{9} - \frac{y^2}{16} = 1 \]

### Step 2: Identify \(a^2\) and \(b^2\)
From the standard form of the hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), we can identify:
\[ a^2 = 9 \quad \text{and} \quad b^2 = 16 \]
Thus, we have:
\[ a = 3 \quad \text{and} \quad b = 4 \]

### Step 3: Use the formula for eccentricity
The eccentricity \(e\) of a hyperbola is given by the formula:
\[ e = \sqrt{1 + \frac{b^2}{a^2}} \]
Substituting the values of \(a^2\) and \(b^2\):
\[ e = \sqrt{1 + \frac{16}{9}} \]

### Step 4: Simplify the expression
Now, we simplify the expression inside the square root:
\[ e = \sqrt{1 + \frac{16}{9}} = \sqrt{\frac{9}{9} + \frac{16}{9}} = \sqrt{\frac{25}{9}} \]

### Step 5: Calculate the final value of eccentricity
Taking the square root gives us:
\[ e = \frac{\sqrt{25}}{\sqrt{9}} = \frac{5}{3} \]

Thus, the value of the eccentricity of the hyperbola is:
\[ \boxed{\frac{5}{3}} \]