The equation of a tangent to the hyperbola `3x^(2)-y^(2)=3`, parallel to the line `y = 2x +4` is

Options

A. `y = 2x +3`
B. `y = 2x+1`
C. `y = 2x+4`
D. `y = 2x+2`

Explanation

To find the equation of the tangent to the hyperbola \(3x^2 - y^2 = 3\) that is parallel to the line \(y = 2x + 4\), we will follow these steps:

### Step 1: Rewrite the hyperbola in standard form
The given equation of the hyperbola is:
\[ 3x^2 - y^2 = 3 \]
Dividing both sides by 3, we get:
\[ \frac{x^2}{1} - \frac{y^2}{3} = 1 \]
This shows that \(a^2 = 1\) and \(b^2 = 3\).

### Step 2: Identify the slope of the given line
The equation of the line is:
\[ y = 2x + 4 \]
From this equation, we can see that the slope \(m\) of the line is 2.

### Step 3: Use the tangent formula for hyperbolas
The formula for the equation of the tangent to the hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) is given by:
\[ y = mx \pm \sqrt{a^2 m^2 - b^2} \]
Substituting \(m = 2\), \(a^2 = 1\), and \(b^2 = 3\) into the formula, we have:
\[ y = 2x \pm \sqrt{1 \cdot 2^2 - 3} \]

### Step 4: Calculate the square root term
Calculating the term inside the square root:
\[ 1 \cdot 2^2 - 3 = 4 - 3 = 1 \]
Thus, we have:
\[ \sqrt{1} = 1 \]

### Step 5: Write the equations of the tangents
Now substituting back into the tangent equation:
\[ y = 2x \pm 1 \]
This gives us two equations for the tangents:
1. \(y = 2x + 1\)
2. \(y = 2x - 1\)

### Final Answer
The equations of the tangents to the hyperbola \(3x^2 - y^2 = 3\) that are parallel to the line \(y = 2x + 4\) are:
\[ y = 2x + 1 \quad \text{and} \quad y = 2x - 1 \]