The eccentricity of the ellipse `(x^(2))/(49) +(y^(2))/(36) = 1 ` is `(sqrt(5))/(3)`.

Answer

To find the eccentricity of the ellipse given by the equation

\[ \frac{x^2}{49} + \frac{y^2}{36} = 1, \]

we will follow these steps:

### Step 1: Identify the values of \(a^2\) and \(b^2\)

The standard form of the ellipse is

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. \]

From the given equation, we can identify:

- \(a^2 = 49\)
- \(b^2 = 36\)

### Step 2: Calculate \(a\) and \(b\)

Now, we will find \(a\) and \(b\):

\[ a = \sqrt{49} = 7, \]
\[ b = \sqrt{36} = 6. \]

### Step 3: Use the formula for eccentricity \(e\)

The formula for the eccentricity \(e\) of an ellipse is given by:

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

### Step 4: Substitute the values of \(a^2\) and \(b^2\)

Now, substituting the values we found:

\[ e = \sqrt{1 - \frac{36}{49}}. \]

### Step 5: Simplify the expression

Calculating the fraction:

\[ \frac{36}{49} = 0.734693877. \]

Now, subtracting from 1:

\[ 1 - \frac{36}{49} = \frac{49 - 36}{49} = \frac{13}{49}. \]

### Step 6: Take the square root

Now, we take the square root:

\[ e = \sqrt{\frac{13}{49}} = \frac{\sqrt{13}}{7}. \]

### Step 7: Compare with the given eccentricity

The problem states that the eccentricity is \(\frac{\sqrt{5}}{3}\).

We need to check if \(\frac{\sqrt{13}}{7}\) is equal to \(\frac{\sqrt{5}}{3}\):

Cross-multiplying gives:

\[ \sqrt{13} \cdot 3 \quad \text{and} \quad \sqrt{5} \cdot 7. \]

Calculating both sides:

- Left side: \(3\sqrt{13}\)
- Right side: \(7\sqrt{5}\)

Since \(3\sqrt{13} \neq 7\sqrt{5}\), the statement that the eccentricity is \(\frac{\sqrt{5}}{3}\) is false.

### Final Conclusion

The eccentricity of the ellipse \(\frac{x^2}{49} + \frac{y^2}{36} = 1\) is \(\frac{\sqrt{13}}{7}\), which is not equal to \(\frac{\sqrt{5}}{3}\).

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