For the ellipse `(x ^(2))/(64) + (y^(2))/(28)=1,` the eccentricity is

Options

A. `3/4`
B. `4/3`
C. `(1)/(sqrt7)`
D. `1/3`

Answer

To find the eccentricity of the ellipse given by the equation

\[ \frac{x^2}{64} + \frac{y^2}{28} = 1, \]

we can follow these steps:

### Step 1: Identify \( a^2 \) and \( b^2 \)
From the equation of the ellipse, we can identify:
- \( a^2 = 64 \)
- \( b^2 = 28 \)

### Step 2: Determine \( a \) and \( b \)
Next, we find \( a \) and \( b \):
- \( a = \sqrt{64} = 8 \)
- \( b = \sqrt{28} = 2\sqrt{7} \)

### Step 3: Check the relationship between \( a \) and \( b \)
Since \( a > b \) (8 > \( 2\sqrt{7} \)), we can proceed to calculate the eccentricity using the formula for eccentricity of an ellipse:

\[ e = \sqrt{1 - \frac{b^2}{a^2}}. \]

### Step 4: Substitute \( a^2 \) and \( b^2 \) into the formula
Substituting the values we found:
\[ e = \sqrt{1 - \frac{28}{64}}. \]

### Step 5: Simplify the fraction
To simplify \( \frac{28}{64} \):
\[ \frac{28}{64} = \frac{7}{16}. \]

### Step 6: Substitute back into the eccentricity formula
Now substitute this back into the equation for \( e \):
\[ e = \sqrt{1 - \frac{7}{16}}. \]

### Step 7: Find a common denominator
To perform the subtraction:
\[ 1 = \frac{16}{16} \quad \text{so} \quad 1 - \frac{7}{16} = \frac{16 - 7}{16} = \frac{9}{16}. \]

### Step 8: Take the square root
Now we take the square root:
\[ e = \sqrt{\frac{9}{16}} = \frac{3}{4}. \]

### Final Answer
Thus, the eccentricity of the ellipse is

\[ \frac{3}{4}. \]

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