Question:
The axis of the parabola \( x^2 + 6x + 4y + 5 = 0 \) is:
The axis of the parabola \( x^2 + 6x + 4y + 5 = 0 \) is:
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For parabolas with only \(x^2\), always complete the square in \(x\).
The axis of symmetry is directly given by \( x = -\frac{D}{2} \) from \( x^2 + Dx \).
Updated On: Apr 30, 2026
- \( x = 0 \)
- \( y = 1 \)
- \( x + 3 = 0 \)
- \( y = 4 \)
- \( y + 2 = 0 \)
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The Correct Option is C
Solution and Explanation
Concept:
A parabola of the form
\[
x^2 + Dx + Ey + F = 0
\]
represents a vertical parabola (since only \(x^2\) is present).
To find its axis:
• Convert the equation into vertex form by completing the square.
• The axis of symmetry is a vertical line passing through the vertex: \( x = \text{constant} \).
Step 1: Group the \(x\)-terms and complete the square. \[ x^2 + 6x + 4y + 5 = 0 \] \[ (x^2 + 6x) + 4y + 5 = 0 \] Complete the square: \[ x^2 + 6x = (x+3)^2 - 9 \] Substitute: \[ (x+3)^2 - 9 + 4y + 5 = 0 \] \[ (x+3)^2 + 4y - 4 = 0 \]
Step 2: Rewrite in standard form. \[ (x+3)^2 = -4(y - 1) \]
Step 3: Identify the axis of symmetry. The vertex form is: \[ (x - h)^2 = 4a(y - k) \] where the axis is: \[ x = h \] Here, \( h = -3 \), so the axis is: \[ x = -3 \] \[ \Rightarrow x + 3 = 0 \]
• Convert the equation into vertex form by completing the square.
• The axis of symmetry is a vertical line passing through the vertex: \( x = \text{constant} \).
Step 1: Group the \(x\)-terms and complete the square. \[ x^2 + 6x + 4y + 5 = 0 \] \[ (x^2 + 6x) + 4y + 5 = 0 \] Complete the square: \[ x^2 + 6x = (x+3)^2 - 9 \] Substitute: \[ (x+3)^2 - 9 + 4y + 5 = 0 \] \[ (x+3)^2 + 4y - 4 = 0 \]
Step 2: Rewrite in standard form. \[ (x+3)^2 = -4(y - 1) \]
Step 3: Identify the axis of symmetry. The vertex form is: \[ (x - h)^2 = 4a(y - k) \] where the axis is: \[ x = h \] Here, \( h = -3 \), so the axis is: \[ x = -3 \] \[ \Rightarrow x + 3 = 0 \]
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