What is the eccentricity of the conic `4x^(2)+9y^(2)=144`?

Options

A. `(sqrt(5))/(3)`
B. `(sqrt(15))/(4)`
C. `(3)/(sqrt(5))`
D. `(2)/(3)`

Answer

To find the eccentricity of the conic given by the equation \( 4x^2 + 9y^2 = 144 \), we will follow these steps:

### Step 1: Rewrite the equation in standard form
We start with the equation:
\[ 4x^2 + 9y^2 = 144 \]
To convert this into standard form, we divide the entire equation by 144:
\[ \frac{4x^2}{144} + \frac{9y^2}{144} = 1 \]
This simplifies to:
\[ \frac{x^2}{36} + \frac{y^2}{16} = 1 \]

### Step 2: Identify the values of \(a^2\) and \(b^2\)
From the standard form of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we can identify:
\[ a^2 = 36 \quad \text{and} \quad b^2 = 16 \]
Thus, we find:
\[ a = \sqrt{36} = 6 \quad \text{and} \quad b = \sqrt{16} = 4 \]

### Step 3: Calculate the eccentricity
The formula for the eccentricity \(e\) of an ellipse is given by:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Substituting the values of \(b^2\) and \(a^2\):
\[ e = \sqrt{1 - \frac{16}{36}} \]
This simplifies to:
\[ e = \sqrt{1 - \frac{4}{9}} = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3} \]

### Conclusion
Thus, the eccentricity of the conic \(4x^2 + 9y^2 = 144\) is:
\[ \frac{\sqrt{5}}{3} \]