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A parabola can be drawn given a focus of left parenthesis, 5, comma, minus, 8, right parenthesis(5,−8) and a directrix of y, equals, 4y=4. What can be said about the parabola?
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GPT-5 mini
The directrix y = 4 is horizontal and the focus is at (5, −8), so the axis is the vertical line x = 5 and the parabola opens downward (focus is below the directrix).
Vertex: midpoint between focus and directrix in the vertical direction:
distance = |−8 − 4| = 12, so vertex is 6 units from each: (5, −2).
Focal parameter p = −6 (signed, since it opens downward). Equation in standard form:
(x − 5)^2 = 4p(y + 2) = −24(y + 2),
or y = −2 − (x − 5)^2/24.
Focus: (5, −8). Directrix: y = 4. Latus rectum length = |4p| = 24, endpoints at (5 ± 12, −8) = (−7, −8) and (17, −8).
Vertex: midpoint between focus and directrix in the vertical direction:
distance = |−8 − 4| = 12, so vertex is 6 units from each: (5, −2).
Focal parameter p = −6 (signed, since it opens downward). Equation in standard form:
(x − 5)^2 = 4p(y + 2) = −24(y + 2),
or y = −2 − (x − 5)^2/24.
Focus: (5, −8). Directrix: y = 4. Latus rectum length = |4p| = 24, endpoints at (5 ± 12, −8) = (−7, −8) and (17, −8).
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