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Which equation represents a parabola with a focus of (0, 4) and a directrix of y = 2?
y = x2 + 3
y = -x2 + 1
y = x2/2 + 3
y = x2/4 + 3
Solution:
The definition of a parabola states that all points on the parabola always have the same distance to the focus and the directrix.
Let A = (x, y) be a point on the parabola.
Focus, F = (0, 4)
Given, directrix y = 2
D = (x, 2) represent the closest point on the directrix
First, find out the distance using the distance formula,
d = √(x2 - x1)2 + (y2 - y1)2
Distance A and F is dAF = √(x - 0)2 + (y - 4)2 = √(x)2 + (y - 4)2
Distance between A and D is dAD = √(x - x)2 + (y - 2)2 = √(y - 2)2
Since these distances must be equal to each other,
⇒ √(x)2 + (y - 4)2 = √(y - 2)2
Squaring both sides,
⇒ (√(x)2 + (y - 4)2)2 = (√(y - 2)2)2
⇒ (x)2 + (y - 4)2 = (y - 2)2
⇒ x2 + y2 - 8y + 16 = y2 - 4y + 4
Grouping of common terms,
⇒ x2 + y2 - y2 - 8y + 4y + 16 - 4 = 0
⇒ x2 - 4y + 12 = 0
⇒ x2 = 4y - 12
⇒ 4y = x2 + 12
⇒ y = 1/4[x2 + 12]
⇒ y = (x2/4) + 3
Therefore, the quadratic function is y = (x2/4) + 3.
Which equation represents a parabola with a focus of (0, 4) and a directrix of y = 2?
Summary:
The equation y = (x2/4) + 3 represents a parabola with a focus of (0, 4) and a directrix of y = 2.
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