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The focus of a parabola is (0, -3) and the directrix is y = 3. What is the equation of the parabola?

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Answer:

The equation of the parabola is
y = -12\cdot x^(2).

Explanation:

Since the directrix is of the form
y = a, the parabola is vertical. The vertex of the parabola (
V(x,y)) is the midpoint between the focus (
F(x,y)) and a point of the directrix (
P(x,y)), that is to say:


V(x,y) = (1)/(2)\cdot F(x,y) + (1)/(2)\cdot P(x,y) (1)

If we know that
F(x,y) = (0, -3) and
P(x,y) = (0, 3), then the coordinates of the vertex of the parabola:


V(x,y) = (1)/(2)\cdot (0, -3) + (1)/(2)\cdot (0, 3)


V(x,y) = (0, 0)

The vertex factor (
p) is the distance of the focus with respect to the vertex:


p = √([F(x,y)-V(x,y)]\,\bullet \,[F(x,y)-V(x,y)]) (2)

If we know that
F(x,y) = (0, -3) and
V(x,y) = (0, 0), then the vertex factor is:


p = \sqrt{0^(2)+(-3)^(2)}


p = -3

The equation of the parabola in the vertex form is described below:


y - k = 4\cdot p \cdot (x-h)^(2) (3)

Where:


x - Independent variable.


y - Dependent variable.


h, k - Coordinates of the vertex.

If we know that
(h,k) = (0, 0) and
p = -3, then the equation of the parabola is
y = -12\cdot x^(2).

The focus of a parabola is (0, -3) and the directrix is y = 3. What is the equation-example-1
User Vishnu Sureshkumar
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