Question

The length of the axis of the conic \(9x^2 + 4y^2 - 6x + 4y + 1 = 0\) are

• A. \(1/2, 9\)
• B. \(3, 2/5\)
• C. \(1, 2/3\)
• D. \(3, 2\)

Hint:

For ellipse, major axis length = \(2b\) if \(b>a\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Rewrite in standard ellipse form.
Step 2: Detailed Explanation:
\(9(x^2 - 2x/3) + 4(y^2 + y) + 1 = 0\)
\(9[(x - 1/3)^2 - 1/9] + 4[(y + 1/2)^2 - 1/4] + 1 = 0\)
\(9(x - 1/3)^2 - 1 + 4(y + 1/2)^2 - 1 + 1 = 0\)
\(9(x - 1/3)^2 + 4(y + 1/2)^2 = 1\)
\(\frac{(x - 1/3)^2}{1/9} + \frac{(y + 1/2)^2}{1/4} = 1\)
\(a = 1/3\), \(b = 1/2 \rightarrow\) major axis = \(2b = 1\), minor axis = \(2a = 2/3\)
Step 3: Final Answer:
Lengths are \(1\) and \(2/3\).