The eccentricity of the hyperbola `3x^(2)-y^(2)=4` is :

Options

A. 1
B. 2
C. `-2`
D. 5

Answer

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

### Step 1: Rewrite the equation in standard form
The standard form of a hyperbola is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
We start with the equation:
\[ 3x^2 - y^2 = 4 \]
To convert this into standard form, divide the entire equation by 4:
\[ \frac{3x^2}{4} - \frac{y^2}{4} = 1 \]

### 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:
\[ \frac{x^2}{\frac{4}{3}} - \frac{y^2}{4} = 1 \]
Thus, we have:
\[ a^2 = \frac{4}{3} \quad \text{and} \quad b^2 = 4 \]

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

### Step 4: Final result
Therefore, the eccentricity \(e\) is:
\[ e = 2 \]

### Summary
The eccentricity of the hyperbola \(3x^2 - y^2 = 4\) is \(2\).

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