How to find equation of parabola with vertex and focus

Asked by admin @ in Mathematics viewed by 314 People

What is the equation of the parabola with a vertex at (-2, 3) and a focus at (-2, 0)? Does it open upward or downward?

a.
(x + 2)883-12-01-00-00_files/i0060000.jpg = 0.4(y - 3); upward
c.
(x + 2)883-12-01-00-00_files/i0060001.jpg = -12(y - 3); downward
b.
(x + 2)883-12-01-00-00_files/i0060002.jpg = 8(y - 3); upward
d.
(x + 2)883-12-01-00-00_files/i0060003.jpg = -8(y - 3); downward

Answered by admin @


Refer to the diagram shown below.

The distance, d, from the focus to a point P (x,y) is equal to the distance from the directrix to P.
Therefore
d² = (6 - y)²
     = (x + 2)² + y²
That is,
36 - 12y + y² = (x + 2)² + y²
12y = -(x + 2)² + 36

The equation of the parabola is
y = -(1/12)(x + 2)² + 3
It may be written as
-12(y - 3) = (x + 2)²

The leading coefficient -1/12 is negative, therefore the curve opens downward.

Answer:
The equation is (x + 2)² = -12(y - 3).
The curve opens downward.









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