Question:
The equation of the latus rectum of the parabola $y^2 + 8x + 4y + 12 = 0$ is
The equation of the latus rectum of the parabola $y^2 + 8x + 4y + 12 = 0$ is
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Latus rectum passes through focus and is perpendicular to axis.
Updated On: Apr 24, 2026
- $x + 3 = 0$
- $y + 3 = 0$
- $x + 1 = 0$
- $y + 2 = 0$
- $x + 2 = 0$
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The Correct Option is A
Solution and Explanation
Concept:
• Standard form: $(y-k)^2 = 4a(x-h)$
• Latus rectum is line through focus parallel to directrix
Step 1: Rewrite equation
\[ y^2 + 4y + 8x + 12 = 0 \] Complete square: \[ (y+2)^2 - 4 + 8x + 12 = 0 \] \[ (y+2)^2 + 8x + 8 = 0 \] \[ (y+2)^2 = -8(x+1) \]
Step 2: Identify parameters
\[ (h,k) = (-1,-2), \quad 4a = -8 \Rightarrow a = -2 \]
Step 3: Find focus
\[ \text{Focus} = (h+a, k) = (-1-2, -2) = (-3,-2) \]
Step 4: Equation of latus rectum
Since axis is horizontal, latus rectum is vertical line: \[ x = -3 \] \[ x + 3 = 0 \] Final Conclusion:
Option (A)
• Standard form: $(y-k)^2 = 4a(x-h)$
• Latus rectum is line through focus parallel to directrix
Step 1: Rewrite equation
\[ y^2 + 4y + 8x + 12 = 0 \] Complete square: \[ (y+2)^2 - 4 + 8x + 12 = 0 \] \[ (y+2)^2 + 8x + 8 = 0 \] \[ (y+2)^2 = -8(x+1) \]
Step 2: Identify parameters
\[ (h,k) = (-1,-2), \quad 4a = -8 \Rightarrow a = -2 \]
Step 3: Find focus
\[ \text{Focus} = (h+a, k) = (-1-2, -2) = (-3,-2) \]
Step 4: Equation of latus rectum
Since axis is horizontal, latus rectum is vertical line: \[ x = -3 \] \[ x + 3 = 0 \] Final Conclusion:
Option (A)
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