The eccentricity of the ellipse `12x^(2)+7y^(2)=84` is equal to

Options

A. `(sqrt(5))/7`
B. `sqrt(5/12)`
C. `(sqrt(5))/12`
D. `5/7`

Answer

To find the eccentricity of the ellipse given by the equation \( 12x^2 + 7y^2 = 84 \), we will follow these steps:

### Step 1: Rewrite the equation in standard form
We start with the equation of the ellipse:
\[ 12x^2 + 7y^2 = 84 \]
To rewrite this in standard form, we divide the entire equation by 84:
\[ \frac{12x^2}{84} + \frac{7y^2}{84} = 1 \]
This simplifies to:
\[ \frac{x^2}{7} + \frac{y^2}{12} = 1 \]

### 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:
\[ a^2 = 7 \quad \text{and} \quad b^2 = 12 \]

### Step 3: Determine \(a\) and \(b\)
Next, we find the values of \(a\) and \(b\):
\[ a = \sqrt{7} \quad \text{and} \quad b = \sqrt{12} \]

### Step 4: Calculate the eccentricity
The eccentricity \(e\) of an ellipse is given by the formula:
\[ e = \sqrt{1 - \frac{a^2}{b^2}} \]
Substituting the values of \(a^2\) and \(b^2\):
\[ e = \sqrt{1 - \frac{7}{12}} \]
This simplifies to:
\[ e = \sqrt{\frac{12 - 7}{12}} = \sqrt{\frac{5}{12}} \]

### Step 5: Final result
Thus, the eccentricity of the ellipse is:
\[ e = \frac{\sqrt{5}}{\sqrt{12}} = \frac{\sqrt{5}}{2\sqrt{3}} = \frac{\sqrt{15}}{6} \]

### Summary
The eccentricity of the ellipse \(12x^2 + 7y^2 = 84\) is \(\frac{\sqrt{5}}{\sqrt{12}}\).