The equation of the tangents to the ellipse `4x^(2)+3y^(2)=5`, which are parallel to the line y=3x+7 are

Options

A. `y=3x+-sqrt((155)/(3))`
B. `y=3x+-sqrt((155)/(12))`
C. `y=3x+-sqrt((95)/(12))`
D. none of these

Explanation

To find the equations of the tangents to the ellipse \(4x^2 + 3y^2 = 5\) that are parallel to the line \(y = 3x + 7\), we can follow these steps:

### Step 1: Convert the equation of the ellipse to standard form
We start with the given equation of the ellipse:
\[ 4x^2 + 3y^2 = 5 \]
To convert this into standard form, we divide the entire equation by 5:
\[ \frac{4x^2}{5} + \frac{3y^2}{5} = 1 \]
This can be rewritten as:
\[ \frac{x^2}{\frac{5}{4}} + \frac{y^2}{\frac{5}{3}} = 1 \]
From this, we identify \(a^2 = \frac{5}{4}\) and \(b^2 = \frac{5}{3}\).

### Step 2: Identify the slope of the tangent line
The line \(y = 3x + 7\) has a slope of \(m = 3\). Since we are looking for tangents to the ellipse that are parallel to this line, the slope of our tangents will also be \(m = 3\).

### Step 3: Use the slope form of the tangent to the ellipse
The slope form of the tangent to the ellipse is given by:
\[ y = mx \pm \sqrt{a^2 m^2 + b^2} \]
Substituting \(m = 3\), \(a^2 = \frac{5}{4}\), and \(b^2 = \frac{5}{3}\) into this formula, we get:
\[ y = 3x \pm \sqrt{\frac{5}{4} \cdot 3^2 + \frac{5}{3}} \]
Calculating \(3^2\):
\[ 3^2 = 9 \]
Thus, we have:
\[ y = 3x \pm \sqrt{\frac{5}{4} \cdot 9 + \frac{5}{3}} \]
Calculating \(\frac{5}{4} \cdot 9\):
\[ \frac{5 \cdot 9}{4} = \frac{45}{4} \]
Now, we need to find a common denominator to add \(\frac{45}{4}\) and \(\frac{5}{3}\). The least common multiple of 4 and 3 is 12. Therefore, we convert both fractions:
\[ \frac{45}{4} = \frac{135}{12}, \quad \frac{5}{3} = \frac{20}{12} \]
Adding these:
\[ \frac{135}{12} + \frac{20}{12} = \frac{155}{12} \]
Now substituting back into the tangent equation:
\[ y = 3x \pm \sqrt{\frac{155}{12}} \]

### Step 4: Final equation of the tangents
Thus, the equations of the tangents to the ellipse that are parallel to the line \(y = 3x + 7\) are:
\[ y = 3x + \sqrt{\frac{155}{12}} \quad \text{and} \quad y = 3x - \sqrt{\frac{155}{12}} \]

### Summary of the solution:
The final answer is:
\[ y = 3x \pm \sqrt{\frac{155}{12}} \]