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Question
Math
Posted 6 months ago

Write the equation for a parabola with a focus at (4,3)(-4,3) and a directrix at y=5y=5.
y=
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Answer from Sia
Posted 6 months ago
Solution by Steps
step 1
Identify the focus and directrix of the parabola. The focus is given as (4,3)(-4, 3) and the directrix is the line y=5y = 5
step 2
Recall the definition of a parabola. A parabola is the set of all points that are equidistant from the focus and the directrix
step 3
Determine the vertex of the parabola. The vertex lies midway between the focus and directrix. Since the directrix is y=5y = 5 and the focus has a yy-coordinate of 33, the vertex yy-coordinate is 5+32=4\frac{5+3}{2} = 4. The xx-coordinate of the vertex is the same as the focus, which is 4-4. Thus, the vertex is (4,4)(-4, 4)
step 4
Write the standard form of the equation for a vertical parabola with vertex (h,k)(h, k): (xh)2=4p(yk)(x-h)^2 = 4p(y-k), where pp is the distance from the vertex to the focus or directrix
step 5
Calculate the value of pp. The distance from the vertex to the focus or directrix is 11, since the vertex is at y=4y = 4 and the directrix is at y=5y = 5
step 6
Substitute the vertex coordinates and the value of pp into the standard form equation. We get (x(4))2=4(1)(y4)(x - (-4))^2 = 4(1)(y - 4)
step 7
Simplify the equation to find the equation of the parabola. (x+4)2=4(y4)(x + 4)^2 = 4(y - 4)
Answer
The equation of the parabola is (x+4)2=4(y4)(x + 4)^2 = 4(y - 4).
Key Concept
Standard form of a parabola's equation
Explanation
The standard form of a parabola's equation is derived from the definition of a parabola, using the distance from the vertex to the focus or directrix to determine the value of pp in the equation (xh)2=4p(yk)(x-h)^2 = 4p(y-k).

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