If the focus is (2, 5) and the directrix is y=3, find the equation of the parabola.
2 answers:
Answer:
y = 1/4 (x - 2)^2 + 4
Step-by-step explanation:
The equation of a parabola is
y = a(x - h)^2 + k
This is for a parabola that opens in the y direction (up or down) If you make a sketch of the focus and directrix you'll be able to see what direction it opens. See image. The parabola kind of curls around the focus and opens away from the directrix. Also, this sketch can help you find the vertex which we need, which is actually a point on the curve and which is exactly half way between the focus and directrix.
The vertex is (h, k) for this problem it is (2, 4). h is 2 and k is 4. You will fill in the 2 and 4 in the parabola formula. Then the last thing you need is the a infront of the paranthesis on the right side of the equation.
Use the formula
a = 1/4p (the p is in the denominator).
p is the distance from the vertex to the focus, so for this question, it is 1. see image.
a = 1/4(1)
a = 1/4
Fill it in the parabola formula.
y = a (x - h)^2 + k
y = 1/4 (x - 2)^2 + 4
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Length: 2w + 59
width: w
diagonal: (2w + 59) + 2 = 2w + 61
Length² + width² = diagonal²
(2w + 59)² + (w)² = (2w + 61)²
(4w² + 118w + 3481) + w² = 4w² + 122w + 3721
5w² + 118w + 3481 = 4w² + 122w + 3721
w² + 118w + 3481 = 122w + 3721
w² - 4w + 3481 = 3721
w² - 4w - 240 = 0
a = 1, b = -4, c = -240
w =
=
=
=
=
=
since width cannot be negative, disregard 1 - 2√61
w = 1 + 2√61 ≈ 16.62
Length: 2w + 59 = 2(1 + 2√61) + 59 = 2 + 4√61 + 59 = 61 + 4√61 ≈ 92.24
Answer: width = 16.62 in, length = 92.24 in