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Find the equation of the parabola with focus at (0, -3) and directrix y = 3.
Find the equation of the parabola with focus at (0, -3) and directrix y = 3.
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Practice Questions
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Q1
Find the equation of the parabola with focus at (0, -3) and directrix y = 3.
x^2 = -12y
x^2 = 12y
y^2 = -12x
y^2 = 12x
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The distance from the focus to the directrix is 6, so p = -3. The equation is x^2 = 4py, which gives x^2 = -12y.
Questions & Step-by-step Solutions
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Q
Q: Find the equation of the parabola with focus at (0, -3) and directrix y = 3.
Solution:
The distance from the focus to the directrix is 6, so p = -3. The equation is x^2 = 4py, which gives x^2 = -12y.
Steps: 6
Show Steps
Step 1: Identify the focus and directrix. The focus is at (0, -3) and the directrix is the line y = 3.
Step 2: Calculate the distance between the focus and the directrix. The distance is 3 - (-3) = 6.
Step 3: Determine the value of p. Since the focus is below the directrix, p is negative. Therefore, p = -3.
Step 4: Use the standard equation of a parabola that opens downwards, which is x^2 = 4py.
Step 5: Substitute p into the equation. Since p = -3, we have x^2 = 4 * (-3) * y.
Step 6: Simplify the equation. This gives us x^2 = -12y.
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