Question:
The equation of the parabola whose vertex and focus are \( (0, 4) \) and \( (0, 2) \) respectively, is:
The equation of the parabola whose vertex and focus are \( (0, 4) \) and \( (0, 2) \) respectively, is:
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For parabolas with vertical axes, use the form \( x^2 = 4py \) where \( p \) is the distance from the vertex to the focus.
Updated On: Apr 22, 2026
- \( y^2 - 8x = 32 \)
- \( y^2 + 8x = 32 \)
- \( x^2 + 8y = 32 \)
- \( x^2 - 8y = 32 \)
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The Correct Option is C
Solution and Explanation
Step 1: General form of the parabola.
The general equation of a parabola with its vertex at \( (h, k) \) and focus at \( (h, k + p) \) is: \[ y^2 = 4px \] For a vertical parabola, the equation will be of the form: \[ x^2 = 4py \] where \( p \) is the distance from the vertex to the focus.
Step 2: Use the given coordinates.
We are given the vertex \( (0, 4) \) and focus \( (0, 2) \), so the distance between the vertex and focus is \( p = 2 \). Thus, the equation of the parabola is: \[ x^2 = 8(y - 4) \]
Step 3: Simplify the equation.
Rearranging this equation: \[ x^2 + 8y = 32 \]
Step 4: Conclusion.
Thus, the equation of the parabola is \( x^2 + 8y = 32 \), which corresponds to option (C).
The general equation of a parabola with its vertex at \( (h, k) \) and focus at \( (h, k + p) \) is: \[ y^2 = 4px \] For a vertical parabola, the equation will be of the form: \[ x^2 = 4py \] where \( p \) is the distance from the vertex to the focus.
Step 2: Use the given coordinates.
We are given the vertex \( (0, 4) \) and focus \( (0, 2) \), so the distance between the vertex and focus is \( p = 2 \). Thus, the equation of the parabola is: \[ x^2 = 8(y - 4) \]
Step 3: Simplify the equation.
Rearranging this equation: \[ x^2 + 8y = 32 \]
Step 4: Conclusion.
Thus, the equation of the parabola is \( x^2 + 8y = 32 \), which corresponds to option (C).
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