Question

The eccentricity of the conic \(x^2 - 4x + 4y^2 = 12\) is

• A. \(\frac{\sqrt{3}}{2}\)
• B. \(\frac{2}{\sqrt{3}}\)
• C. \(\sqrt{3}\)
• D. None of these

Hint:

For ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) with \(a>b\), \(e = \sqrt{1 - b^2/a^2}\).

Answer 1

The Correct Option is A

To find the eccentricity of the conic given by the equation \(x^2 - 4x + 4y^2 = 12\), we follow these steps:

1. First, rewrite the equation in the standard form of a conic. The given equation is:
2. Complete the square for the \(x\) terms:
   • The expression \(x^2 - 4x\) can be completed by adding and subtracting 4: \((x^2 - 4x + 4 - 4)\).
   • This becomes: \((x - 2)^2 - 4\).
3. The equation now is:
4. Divide the entire equation by 16 to express it in the standard form:
5. This is the standard form of an ellipse: \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\), where \(a^2 = 16\) and \(b^2 = 4\).
   • Here, \(a^2 > b^2\), indicating it is a horizontal ellipse.
6. The formula for eccentricity \(e\) of an ellipse is:
7. Calculate \(a\) and \(b\):
   • \(a = \sqrt{16} = 4\)
   • \(b = \sqrt{4} = 2\)
8. Substitute the values into the eccentricity formula:
9. Thus, the eccentricity of the given conic is:

The correct answer is therefore: \(\frac{\sqrt{3}}{2}\). Other options are incorrect since they do not match the calculated value of the eccentricity.