Question

The axis of the parabola \(x^{2} + 6x + 4y + 5 = 0\) is

• A. \(x = 0\)
• B. \(y = 1\)
• C. \(x + 3 = 0\)
• D. \(y = 4\)
• E. \(y + 2 = 0\)

Hint:

A quick way to find the axis of a parabola of the form \(x^2 + Ax + By + C = 0\) is to differentiate partially with respect to \(x\): \(2x + A = 0\). Here, \(2x + 6 = 0 \implies x + 3 = 0\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The axis of a parabola is the line of symmetry that passes through the vertex and focus. For a parabola where the \(x\) term is squared, the axis is a vertical line.

Step 2: Key Formula or Approach:
Complete the square for the \(x\) terms to bring the equation into the standard form: \[ (x - h)^2 = 4a(y - k) \] The axis is then given by \(x - h = 0\).

Step 3: Detailed Explanation:
1. Start with \(x^2 + 6x = -4y - 5\).
2. Complete the square by adding \((6/2)^2 = 9\) to both sides: \[ x^2 + 6x + 9 = -4y - 5 + 9 \] \[ (x + 3)^2 = -4y + 4 \]
3. Factor the right side: \((x + 3)^2 = -4(y - 1)\).
4. The equation is in the form \((x-h)^2 = 4a(y-k)\) where \(h = -3\). The axis of symmetry is the line \(x = h\), which is \(x = -3\) or \(x + 3 = 0\).

Step 4: Final Answer
The axis of the parabola is \(x + 3 = 0\).