Question

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

Answer 1

Answer:

The equation of the parabola is $y = -12\cdot x^{2}$.

Step-by-step 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:

$V(x,y) = \frac{1}{2}\cdot F(x,y) + \frac{1}{2}\cdot P(x,y)$ (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:

$V(x,y) = \frac{1}{2}\cdot (0, -3) + \frac{1}{2}\cdot (0, 3)$

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

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

$p = \sqrt{[F(x,y)-V(x,y)]\,\bullet \,[F(x,y)-V(x,y)]}$ (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 $y = -12\cdot x^{2}$.