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

Explanation

To find the range of \(\frac{144}{x} + \frac{25}{y}\) given the equation \(\frac{x^2}{144} - \frac{y^2}{25} = 1\), we can follow these steps:

### Step 1: Rewrite the given equation
The equation can be rewritten in a more recognizable form:
\[ \frac{x^2}{12^2} - \frac{y^2}{5^2} = 1 \]
This represents a hyperbola.

### Step 2: Parameterize \(x\) and \(y\)
To express \(x\) and \(y\) in terms of a parameter, we can use the following substitutions:
\[ x = 12 \sec \theta \quad \text{and} \quad y = 5 \tan \theta \]
This substitution is valid because it satisfies the hyperbola equation.

### Step 3: Substitute \(x\) and \(y\) into the expression
Now we substitute \(x\) and \(y\) into the expression \(\frac{144}{x} + \frac{25}{y}\):
\[ \frac{144}{x} = \frac{144}{12 \sec \theta} = 12 \cos \theta \]
\[ \frac{25}{y} = \frac{25}{5 \tan \theta} = \frac{5}{\tan \theta} = 5 \cot \theta \]
Thus, we have:
\[ \frac{144}{x} + \frac{25}{y} = 12 \cos \theta + 5 \cot \theta \]

### Step 4: Express \(\cot \theta\) in terms of \(\sin \theta\) and \(\cos \theta\)
Recall that \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). Therefore, we can rewrite the expression:
\[ 12 \cos \theta + 5 \frac{\cos \theta}{\sin \theta} = \cos \theta \left(12 + \frac{5}{\sin \theta}\right) \]

### Step 5: Find the range of the expression
To find the range of \(12 \cos \theta + 5 \cot \theta\), we can use the Cauchy-Schwarz inequality or the method of finding extrema. The expression can be maximized and minimized using the following:
\[ R = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]
Thus, the range of \(12 \cos \theta + 5 \sin \theta\) is:
\[ [-13, 13] \]

### Conclusion
Therefore, the range of \(\frac{144}{x} + \frac{25}{y}\) is:
\[ [-13, 13] \]

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