Question
Derive the equation of a parabola given the focus is at (3,4) and the directrix is y=0 . Fill in the missing values of the equation in standard form.(1 point)y=x2−34x+258
Answers
Answered by
GPT-4o mini
To derive the equation of a parabola given its focus and directrix, we first need to identify the type of parabola based on the given information.
**Focus**: (3, 4)
**Directrix**: \(y = 0\)
Since the focus is above the directrix, the parabola opens upwards. The standard form of a parabola that opens upwards is given by:
\[
(x - h)^2 = 4p(y - k)
\]
Where:
- \((h, k)\) is the vertex of the parabola,
- \(p\) is the distance from the vertex to the focus (or directrix).
**Step 1: Find the vertex**
The vertex of the parabola is halfway between the focus and the directrix. The directrix is a horizontal line, so we can find the y-coordinate of the vertex by taking the average of the y-coordinate of the focus and the directrix:
- \(y_{\text{vertex}} = \frac{y_{\text{focus}} + y_{\text{directrix}}}{2} = \frac{4 + 0}{2} = 2\)
The x-coordinate of the vertex is the same as the x-coordinate of the focus since the vertex lies directly beneath the focus:
- \(x_{\text{vertex}} = 3\)
So the vertex \((h, k)\) is at:
\[
(h, k) = (3, 2)
\]
**Step 2: Find \(p\)**
The distance \(p\) is the distance from the vertex to the focus (or the directrix). This can be calculated as:
\[
p = y_{\text{focus}} - y_{\text{vertex}} = 4 - 2 = 2
\]
**Step 3: Plug into the standard formula**
Thus, substituting \(h\), \(k\), and \(p\) into the standard form:
1. \(h = 3\)
2. \(k = 2\)
3. \(4p = 4 \cdot 2 = 8\)
The equation in standard form becomes:
\[
(x - 3)^2 = 8(y - 2)
\]
**Step 4: Expand to standard quadratic form**
We can rearrange this into the form \(y = ax^2 + bx + c\). To do this, we can expand and solve for \(y\):
\[
(x - 3)^2 = 8(y - 2)
\]
Expanding the left side:
\[
x^2 - 6x + 9 = 8(y - 2)
\]
Distributing the 8 on the right side:
\[
x^2 - 6x + 9 = 8y - 16
\]
Adding 16 to both sides:
\[
x^2 - 6x + 25 = 8y
\]
Dividing everything by 8 to isolate \(y\):
\[
y = \frac{1}{8}x^2 - \frac{3}{4}x + \frac{25}{8}
\]
This gives us the parabola's equation:
\[
y = \frac{1}{8}x^2 - \frac{3}{4}x + \frac{25}{8}
\]
In the provided equation format \(y = x^2 - 34x + 258\), it does not directly match the derived equation. The values were likely intended to form a specific transformation or scalar multiplication of the results from a standard \(x^2\) form, but the generated equation from the defined focus and directrix is accurately expressed as:
\[
y = \frac{1}{8}(x^2 - 6x + 25)
\]
If you require adjustments to fit a specific requested format or numerically corresponding values, please clarify the requirements.
**Focus**: (3, 4)
**Directrix**: \(y = 0\)
Since the focus is above the directrix, the parabola opens upwards. The standard form of a parabola that opens upwards is given by:
\[
(x - h)^2 = 4p(y - k)
\]
Where:
- \((h, k)\) is the vertex of the parabola,
- \(p\) is the distance from the vertex to the focus (or directrix).
**Step 1: Find the vertex**
The vertex of the parabola is halfway between the focus and the directrix. The directrix is a horizontal line, so we can find the y-coordinate of the vertex by taking the average of the y-coordinate of the focus and the directrix:
- \(y_{\text{vertex}} = \frac{y_{\text{focus}} + y_{\text{directrix}}}{2} = \frac{4 + 0}{2} = 2\)
The x-coordinate of the vertex is the same as the x-coordinate of the focus since the vertex lies directly beneath the focus:
- \(x_{\text{vertex}} = 3\)
So the vertex \((h, k)\) is at:
\[
(h, k) = (3, 2)
\]
**Step 2: Find \(p\)**
The distance \(p\) is the distance from the vertex to the focus (or the directrix). This can be calculated as:
\[
p = y_{\text{focus}} - y_{\text{vertex}} = 4 - 2 = 2
\]
**Step 3: Plug into the standard formula**
Thus, substituting \(h\), \(k\), and \(p\) into the standard form:
1. \(h = 3\)
2. \(k = 2\)
3. \(4p = 4 \cdot 2 = 8\)
The equation in standard form becomes:
\[
(x - 3)^2 = 8(y - 2)
\]
**Step 4: Expand to standard quadratic form**
We can rearrange this into the form \(y = ax^2 + bx + c\). To do this, we can expand and solve for \(y\):
\[
(x - 3)^2 = 8(y - 2)
\]
Expanding the left side:
\[
x^2 - 6x + 9 = 8(y - 2)
\]
Distributing the 8 on the right side:
\[
x^2 - 6x + 9 = 8y - 16
\]
Adding 16 to both sides:
\[
x^2 - 6x + 25 = 8y
\]
Dividing everything by 8 to isolate \(y\):
\[
y = \frac{1}{8}x^2 - \frac{3}{4}x + \frac{25}{8}
\]
This gives us the parabola's equation:
\[
y = \frac{1}{8}x^2 - \frac{3}{4}x + \frac{25}{8}
\]
In the provided equation format \(y = x^2 - 34x + 258\), it does not directly match the derived equation. The values were likely intended to form a specific transformation or scalar multiplication of the results from a standard \(x^2\) form, but the generated equation from the defined focus and directrix is accurately expressed as:
\[
y = \frac{1}{8}(x^2 - 6x + 25)
\]
If you require adjustments to fit a specific requested format or numerically corresponding values, please clarify the requirements.
There are no human answers yet. A form for humans to post answers is coming very soon!