Question

Derive the equation of a parabola given the focus is at (3,4)
(
3
,
4
)
and the directrix is y=0
𝑦
=
0
. Fill in the missing values of the equation in standard form.

Answer 1

To derive the equation of a parabola given the focus and the directrix, we start with the definitions:

1. The **focus** of the parabola is a point, here given as \( (3, 4) \).
2. The **directrix** is a line, in this case given as \( y = 0 \).

The general form of a parabola can be derived from the property that any point \( (x, y) \) on the parabola is equidistant from the focus and the directrix.

### Step 1: Calculate the Distance to the Focus
The distance from a point \( (x, y) \) to the focus \( (3, 4) \) is given by:
\[
d_{\text{focus}} = \sqrt{(x - 3)^2 + (y - 4)^2}
\]

### Step 2: Calculate the Distance to the Directrix
The distance from the point \( (x, y) \) to the directrix \( y = 0 \) (which is the line where \( y \) equals 0) is:
\[
d_{\text{directrix}} = |y - 0| = |y|
\]

### Step 3: Set Up the Equation
According to the definition of a parabola:
\[
d_{\text{focus}} = d_{\text{directrix}}
\]

This leads to the equation:
\[
\sqrt{(x - 3)^2 + (y - 4)^2} = |y|
\]

### Step 4: Eliminate the Square Root
Square both sides to eliminate the square root:
\[
(x - 3)^2 + (y - 4)^2 = y^2
\]

### Step 5: Simplify the Equation
Expand the left-hand side:
\[
(x - 3)^2 + (y^2 - 8y + 16) = y^2
\]

Now, subtract \( y^2 \) from both sides:
\[
(x - 3)^2 - 8y + 16 = 0
\]

### Step 6: Rearrange to Standard Form
Isolate \( y \):
\[
-8y = -(x - 3)^2 - 16
\]
\[
8y = (x - 3)^2 + 16
\]
\[
y = \frac{1}{8}(x - 3)^2 + 2
\]

### Result
Thus, the equation of the parabola in standard form (vertex form) is:
\[
y = \frac{1}{8}(x - 3)^2 + 2
\]

This indicates a parabola that opens upwards, with vertex at \( (3, 2) \), focus at \( (3, 4) \), and directrix \( y = 0 \).