Find the equation of parabola given focus and directrix.

Steps to Solve

1. Identify the coordinates of the focus and the equation of the directrix

Let the focus be at the point $(h, k)$ and the directrix be given by the line $y = d$ (assuming a horizontal directrix for simplicity). Depending on the specific focus and directrix provided, these coordinates and equation will change.

2. Use the definition of a parabola

A parabola is defined as the set of all points $(x, y)$ that are equidistant from the focus and the directrix.

The distance from a point $(x, y)$ to the focus $(h, k)$ is given by the formula: $$ \sqrt{(x - h)^2 + (y - k)^2} $$

The distance from the point $(x, y)$ to the directrix $y = d$ is given by the absolute value of the difference between the y-coordinate of the point and the directrix: $$ |y - d| $$

3. Equate the distances

Set the distances equal to each other since any point on the parabola satisfies this condition: $$ \sqrt{(x - h)^2 + (y - k)^2} = |y - d| $$

4. Square both sides to eliminate the square root

Squaring both sides gives: $$ (x - h)^2 + (y - k)^2 = (y - d)^2 $$

5. Expand and simplify the equation

Now, expand both sides: $$ (x - h)^2 + (y^2 - 2ky + k^2) = (y^2 - 2dy + d^2) $$

Next, combine like terms and get all terms on one side: $$ (x - h)^2 + k^2 - d^2 = 2y(d - k) $$

6. Rearrange to standard form

Finally, we rearrange the equation to isolate $y$: $$ y = \frac{(x - h)^2 + k^2 - d^2}{2(d - k)} $$

This is the equation of the parabola in vertex form based on the given focus and directrix.