If `(x^(2))/(144)-(y^(2))/(25)=1`. Find the range of `(144)/(x)+(25)/(y)`.

Explanation

To solve the problem, we need to find the range of the expression \(\frac{144}{x} + \frac{25}{y}\) given the hyperbola defined by the equation \(\frac{x^2}{144} - \frac{y^2}{25} = 1\).

### Step-by-Step Solution:

1. **Rewrite the Hyperbola Equation**:
We start by rewriting the given hyperbola equation:
\[ \frac{x^2}{144} - \frac{y^2}{25} = 1 \]
This can be expressed in terms of squares:
\[ \left(\frac{x}{12}\right)^2 - \left(\frac{y}{5}\right)^2 = 1 \]

2. **Parametric Representation**:
For hyperbolas, we can use the parametric equations:
\[ x = 12 \sec(\theta) \quad \text{and} \quad y = 5 \tan(\theta) \]
where \(\theta\) is a parameter.

3. **Substituting into the Expression**:
Now, we substitute these parametric representations into the expression we want to find the range of:
\[ \frac{144}{x} + \frac{25}{y} = \frac{144}{12 \sec(\theta)} + \frac{25}{5 \tan(\theta)} \]
Simplifying this gives:
\[ = \frac{12}{\sec(\theta)} + \frac{5}{\tan(\theta)} = 12 \cos(\theta) + 5 \cot(\theta) \]

4. **Expressing Cotangent**:
Recall that \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\), so we can rewrite the expression as:
\[ 12 \cos(\theta) + 5 \frac{\cos(\theta)}{\sin(\theta)} = \cos(\theta) \left(12 + \frac{5}{\sin(\theta)}\right) \]

5. **Finding Maximum and Minimum Values**:
To find the maximum and minimum values of \(12 \cos(\theta) + 5 \sin(\theta)\), we can use the formula for the maximum value of \(a \cos(\theta) + b \sin(\theta)\):
\[ \sqrt{a^2 + b^2} \]
Here, \(a = 12\) and \(b = 5\):
\[ \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]
Therefore, the maximum value is \(13\) and the minimum value is \(-13\).

6. **Conclusion**:
The range of the expression \(\frac{144}{x} + \frac{25}{y}\) is:
\[ [-13, 13] \]

### Final Answer:
The range of \(\frac{144}{x} + \frac{25}{y}\) is \([-13, 13]\).