The eccentricity of ellipse `4(x-2y+1)^2 +9(2x+y+2)^2=180` is

Options

A. `(sqrt(5))/(3)`
B. `(sqrt(3))/(4)`
C. `(1)/(2)`
D. `(1)/(4)`

Answer

To find the eccentricity of the ellipse given by the equation \(4(x - 2y + 1)^2 + 9(2x + y + 2)^2 = 180\), we will follow these steps:

### Step 1: Divide the entire equation by 180
We start by simplifying the given equation:
\[ \frac{4(x - 2y + 1)^2}{180} + \frac{9(2x + y + 2)^2}{180} = 1 \]
This simplifies to:
\[ \frac{(x - 2y + 1)^2}{45} + \frac{(2x + y + 2)^2}{20} = 1 \]

### Step 2: Identify the values of \(a^2\) and \(b^2\)
From the standard form of the ellipse equation:
\[ \frac{(x - 2y + 1)^2}{a^2} + \frac{(2x + y + 2)^2}{b^2} = 1 \]
we can identify:
- \(a^2 = 45\)
- \(b^2 = 20\)

### 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 \(a^2\) and \(b^2\):
\[ e = \sqrt{1 - \frac{20}{45}} = \sqrt{1 - \frac{4}{9}} = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3} \]

### Final Answer
The eccentricity of the ellipse is:
\[ \frac{\sqrt{5}}{3} \]

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