Find the asymptotes of the hyperbola `4x^(2)-9y^(2)=36`

Answer

To find the asymptotes of the hyperbola given by the equation \(4x^2 - 9y^2 = 36\), we will follow these steps:

### Step 1: Rewrite the equation in standard form
We start with the equation of the hyperbola:
\[ 4x^2 - 9y^2 = 36 \]
To rewrite this in standard form, we divide both sides by 36:
\[ \frac{4x^2}{36} - \frac{9y^2}{36} = 1 \]
This simplifies to:
\[ \frac{x^2}{9} - \frac{y^2}{4} = 1 \]

### Step 2: Identify the standard form parameters
The standard form of a hyperbola is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
From our equation, we can identify:
- \(a^2 = 9\) which gives \(a = 3\)
- \(b^2 = 4\) which gives \(b = 2\)

### Step 3: Write the equations of the asymptotes
The equations of the asymptotes for a hyperbola in standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) are given by:
\[ y = \pm \frac{b}{a} x \]
Substituting the values of \(a\) and \(b\):
\[ y = \pm \frac{2}{3} x \]

### Final Result
Thus, the equations of the asymptotes of the hyperbola \(4x^2 - 9y^2 = 36\) are:
\[ y = \frac{2}{3} x \quad \text{and} \quad y = -\frac{2}{3} x \]

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