The equation of the latusrectum of the hyperbola `3y^(2)-4x^(2)=12` are

Options

A. `y=pm sqrt(11)`
B. `y=pm sqrt3`
C. `y=pm sqrt7`
D. `y=pm sqrt5`

Answer

To find the equations of the latus rectum of the hyperbola given by the equation \(3y^2 - 4x^2 = 12\), we will follow these steps:

### Step 1: Rewrite the equation in standard form
We start with the equation of the hyperbola:
\[ 3y^2 - 4x^2 = 12 \]
To convert it into standard form, we divide both sides by 12:
\[ \frac{3y^2}{12} - \frac{4x^2}{12} = 1 \]
This simplifies to:
\[ \frac{y^2}{4} - \frac{x^2}{3} = 1 \]

### Step 2: Identify \(a^2\) and \(b^2\)
From the standard form of the hyperbola \(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\), we can identify:
\[ b^2 = 4 \quad \text{and} \quad a^2 = 3 \]

### Step 3: Calculate the eccentricity \(e\)
The eccentricity \(e\) of the hyperbola is given by the formula:
\[ e = \sqrt{1 + \frac{a^2}{b^2}} \]
Substituting the values of \(a^2\) and \(b^2\):
\[ e = \sqrt{1 + \frac{3}{4}} = \sqrt{\frac{7}{4}} = \frac{\sqrt{7}}{2} \]

### Step 4: Write the equations of the latus rectum
The equations of the latus rectum for the hyperbola are given by:
\[ y = \pm \frac{b \cdot e}{a} \quad \text{(for hyperbolas of the form } \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\text{)} \]
Substituting the values of \(b\) and \(e\):
\[ y = \pm b \cdot e = \pm 2 \cdot \frac{\sqrt{7}}{2} = \pm \sqrt{7} \]

### Final Answer
Thus, the equations of the latus rectum of the hyperbola are:
\[ y = \pm \sqrt{7} \]

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