2. Find the eccentricity of the hyperbola $$ 9 x^{2}-16 y^{2}=144 $$.

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To find the eccentricity of the hyperbola given by the equation $$ 9 x^{2} - 16 y^{2} = 144 $$, we first need to rewrite the equation in standard form.

1. **Rewrite the equation**: Divide the entire equation by 144 to get it in the standard form of a hyperbola.

$$
   \frac{9 x^{2}}{144} - \frac{16 y^{2}}{144} = 1
   $$

This simplifies to:

$$
   \frac{x^{2}}{16} - \frac{y^{2}}{9} = 1
   $$

2. **Identify $$ a^{2} $$ and $$ b^{2} $$**: From the standard form $$ \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 $$, we can identify:

- $$ a^{2} = 16 $$  (thus $$ a = 4 $$)
   - $$ b^{2} = 9 $$   (thus $$ b = 3 $$)

3. **Calculate the eccentricity $$ e $$**: The formula for the eccentricity of a hyperbola is given by:

$$
   e = \sqrt{1 + \frac{b^{2}}{a^{2}}}
   $$

Substituting the values of $$ a^{2} $$ and $$ b^{2} $$:

$$
   e = \sqrt{1 + \frac{9}{16}}
   $$

Now, let's calculate this expression.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{9}{16}$$
Calculate or simplify the expression $$ 1 + 0.5625 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$1+0.5625$$
- step1: Add the numbers:
  $$1.5625$$
Calculate or simplify the expression $$ \frac{25}{16} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{25}{16}$$
Calculate or simplify the expression $$ \sqrt{\frac{25}{16}} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt{\frac{25}{16}}$$
- step1: Use the properties of radicals:
  $$\frac{\sqrt{25}}{\sqrt{16}}$$
- step2: Simplify the expression:
  $$\frac{5}{4}$$
Now, we can summarize the calculations:

1. We found that $$ \frac{9}{16} = 0.5625 $$.
2. Adding 1 gives us $$ 1 + 0.5625 = \frac{25}{16} $$ or $$ 1.5625 $$.
3. Finally, taking the square root of $$ \frac{25}{16} $$ gives us $$ \frac{5}{4} $$ or $$ 1.25 $$.

Thus, the eccentricity $$ e $$ of the hyperbola $$ 9 x^{2} - 16 y^{2} = 144 $$ is:

$$
e = \frac{5}{4} \quad \text{or} \quad 1.25
$$
The eccentricity of the hyperbola $$ 9 x^{2} - 16 y^{2} = 144 $$ is $$ \frac{5}{4} $$ or 1.25.

Find the eccentricity of the hyperbola .

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