The equation to the pair of asymptotes of the hyperbola `x^(2)/9-y^(2)/5=1` is

Options

A. `x^(2)/9-y^(2)/5+1=0`
B. `x^(2)/9-y^(2)/5=2`
C. `5x^(2)-9y^(2)=0`
D. `9x^(2)-5y^2=0`

Answer

To find the equations of the pair of asymptotes for the hyperbola given by the equation:

\[ \frac{x^2}{9} - \frac{y^2}{5} = 1 \]

we can follow these steps:

### Step 1: Identify the standard form of the hyperbola
The standard form of a hyperbola is given by:

\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

From the given equation, we can identify \(a^2 = 9\) and \(b^2 = 5\). Thus, we have:

\[ a = 3 \quad \text{and} \quad b = \sqrt{5} \]

### Step 2: Write the equations of the asymptotes
For a hyperbola of the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the equations of the asymptotes are given by:

\[ y = \pm \frac{b}{a} x \]

### Step 3: Substitute the values of \(a\) and \(b\)
Substituting the values of \(a\) and \(b\) into the equations of the asymptotes:

\[ y = \pm \frac{\sqrt{5}}{3} x \]

### Step 4: Rearranging the equations
We can rearrange these equations to express them in standard form:

1. \(y = \frac{\sqrt{5}}{3} x\) can be rewritten as:
\[ \sqrt{5} x - 3y = 0 \]

2. \(y = -\frac{\sqrt{5}}{3} x\) can be rewritten as:
\[ \sqrt{5} x + 3y = 0 \]

### Step 5: Combine the equations
The equations of the asymptotes can be combined into one equation:

\[ \sqrt{5} x - 3y = 0 \quad \text{and} \quad \sqrt{5} x + 3y = 0 \]

This can be expressed as:

\[ \sqrt{5} x^2 - 9y^2 = 0 \]

### Final Answer
Thus, the equations of the pair of asymptotes of the hyperbola are:

\[ \sqrt{5} x - 3y = 0 \quad \text{and} \quad \sqrt{5} x + 3y = 0 \]

or equivalently,

\[ 5x^2 - 9y^2 = 0 \]