Question

A parabola can be drawn given a focus of (8, 6) and a directrix of x=4 Write the equation of the parabola in any form. Answer:

Answer 1

Answer:

The equation of the parabola in vertex form is $x-6 = \frac{1}{16}\cdot (y-6)^{2}$.

Step-by-step explanation:

Since the directrix is of the $x = a$, then the equation of the parabola in vertex form is of the form:

$x-h = \frac{1}{4\cdot p}\cdot (y-k)^{2}$ (1)

Where:

$x$ - Dependent variable.

$y$ - Independent variable.

$p$ - Least distance between focus and directrix.

$h$, $k$ - Coordinate of the vertex.

The least distance between focus and directrix is determined by Pythagorean Theorem:

$p = \sqrt{(8-4)^{2}+(6-6)^{2}}$

$p = 4$

Now, we determine the location of the vertex by the following vectorial formula:

$(h,k) = F(x,y) - (0.5\cdot p, 0)$ (2)

If we know that $p = 4$ and $F(x,y) = (8,6)$, then the location of the vertex is:

$(h,k) = (8,6) -(2, 0)$

$(h,k) = (6,6)$

And the equation of the parabola in vertex form is $x-6 = \frac{1}{16}\cdot (y-6)^{2}$.