The equations of the asymptotes of the graph of `4x^(2)-9y^(2) = 36` are

Options

A. y=x and y=-x
B. y=0 and x=0
C. `y=2/3x and y=-2/3x`
D. `y=3/2x and y=-3/2x`

Explanation

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

### Step 1: Rewrite the equation in standard form
We start with the equation:
\[ 4x^2 - 9y^2 = 36 \]
To convert this into 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 \(a^2\) and \(b^2\)
From the standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), we can identify:
- \(a^2 = 9\) which gives \(a = 3\)
- \(b^2 = 4\) which gives \(b = 2\)

### Step 3: Determine the orientation of the hyperbola
Since \(a > b\), this hyperbola is oriented along the x-axis.

### Step 4: Write the equations of the asymptotes
For a hyperbola that is oriented along the x-axis, the equations of the asymptotes are given by:
\[ y = \pm \frac{b}{a} x \]
Substituting the values of \(b\) and \(a\):
\[ y = \pm \frac{2}{3} x \]

### Final Answer
Thus, the equations of the asymptotes are:
\[ y = \frac{2}{3} x \quad \text{and} \quad y = -\frac{2}{3} x \]

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