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

To find the lengths of the axes of the conic given by the equation \(9x^2 + 4y^2 - 6x + 4y + 1 = 0\), we need to first determine the type of conic section it represents and then compute the lengths of the major and minor axes.

We start by rewriting the given equation:

\(9x^2 + 4y^2 - 6x + 4y + 1 = 0\)

We can write this equation in the standard form of a conic section by completing the square for the quadratic terms. First, for the \(x\)-terms:

• Take \(9x^2 - 6x\): Factor out 9 to get \(9(x^2 - \frac{2}{3}x)\). Complete the square inside the bracket:
  • \(x^2 - \frac{2}{3}x\) becomes \((x - \frac{1}{3})^2 - \frac{1}{9}\) when completed.

Next, simplify the \(y\)-terms:

• Take \(4y^2 + 4y\): Factor out 4 to get \(4(y^2 + y)\). Complete the square inside the bracket:
  • \(y^2 + y\) becomes \((y + \frac{1}{2})^2 - \frac{1}{4}\) when completed.

Substitute these into the original equation:

• \(9\left((x - \frac{1}{3})^2 - \frac{1}{9}\right) + 4\left((y + \frac{1}{2})^2 - \frac{1}{4}\right) + 1 = 0\)

Simplify this:

• Expand and simplify: \(9(x - \frac{1}{3})^2 + 4(y + \frac{1}{2})^2 = \frac{1}{9} \cdot 9 + \frac{1}{4} \cdot 4 - 1\)
• This becomes: \(9(x - \frac{1}{3})^2 + 4(y + \frac{1}{2})^2 = 1\)

Divide the entire equation by 1 to get a familiar form:

• \(\frac{(x - \frac{1}{3})^2}{\frac{1}{9}} + \frac{(y + \frac{1}{2})^2}{\frac{1}{4}} = 1\)

This is the standard form of an ellipse centered at \((\frac{1}{3}, -\frac{1}{2})\), with semi-major axis \(b = \frac{1}{2}\) (since it is under x-term) and semi-minor axis \(a = \frac{2}{3}\) (since it is under y-term).

Thus, the lengths of the axes of the ellipse are twice their respective semi-axes:

• Major Axis: \(2b = 1\)
• Minor Axis: \(2a = \frac{2}{3}\)

Therefore, the lengths of the axes of the ellipse are \(1, \frac{2}{3}\).

The correct answer is: \(1, 2/3\)