The eccentricity of the ellipse `x^(2)/9+y^(2)/16=1` is

Options

A. `7/16`
B. `5/4`
C. `sqrt(7)/(4)`
D. `sqrt(7)/(2)`

Answer

To find the eccentricity of the ellipse given by the equation \( \frac{x^2}{9} + \frac{y^2}{16} = 1 \), we will follow these steps:

### Step 1: Identify the values of \( a^2 \) and \( b^2 \)
The standard form of the ellipse is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
From the given equation \( \frac{x^2}{9} + \frac{y^2}{16} = 1 \), we can identify:
- \( a^2 = 9 \)
- \( b^2 = 16 \)

### Step 2: Determine \( a \) and \( b \)
Next, we find \( a \) and \( b \) by taking the square roots:
\[ a = \sqrt{9} = 3 \]
\[ b = \sqrt{16} = 4 \]

### Step 3: Check which is the major axis
Since \( b > a \) (4 > 3), the major axis is along the y-axis.

### Step 4: Use the formula for eccentricity
The formula for the eccentricity \( e \) of an ellipse where \( b \) is the semi-major axis and \( a \) is the semi-minor axis is given by:
\[ e = \sqrt{1 - \frac{a^2}{b^2}} \]
Substituting the values of \( a^2 \) and \( b^2 \):
\[ e = \sqrt{1 - \frac{9}{16}} \]

### Step 5: Simplify the expression
Now, simplify the expression inside the square root:
\[ e = \sqrt{1 - \frac{9}{16}} = \sqrt{\frac{16 - 9}{16}} = \sqrt{\frac{7}{16}} \]

### Step 6: Final calculation
Now, we can simplify further:
\[ e = \frac{\sqrt{7}}{\sqrt{16}} = \frac{\sqrt{7}}{4} \]

### Conclusion
Thus, the eccentricity of the ellipse is:
\[ \boxed{\frac{\sqrt{7}}{4}} \]

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