9. Find the standard form of the equation of the parabola with a focus at (-3, 4) and directrix y = 2.
9. Find the standard form of the equation of the parabola with a focus at (-3, 4) and directrix y = 2.
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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find the standard form of the equation of the parabola with a focus at -3,4 and directrix at y=2
Transcribed Image Text:The image is of a Cartesian coordinate graph. The graph is a two-dimensional grid marked with both horizontal (x-axis) and vertical (y-axis) lines. Each axis is labeled with numbers ranging from -10 to 10.
- **X-Axis (Horizontal):**
- The center is labeled at 0.
- Positive numbers increase to the right, going up to 10.
- Negative numbers decrease to the left, going down to -10.
- **Y-Axis (Vertical):**
- The center is labeled at 0.
- Positive numbers increase upwards, going up to 10.
- Negative numbers decrease downwards, going down to -10.
The graph consists of a grid made up of smaller squares, which allows for precise plotting of points. Each square represents one unit on both axes. The bold lines at every 5 units help in identifying larger values easily.
![**Problem:**
Find the standard form of the equation of the parabola with a focus at (-3, 4) and directrix \( y = 2 \).
**Explanation:**
The problem includes an empty coordinate grid with axes labeled.
To solve the problem, consider the following:
1. **Key points:**
- Focus: \((-3, 4)\)
- Directrix: \(y = 2\)
2. **Determine the Vertex:**
- The vertex of a parabola is equidistant from the focus and the directrix. The distance between the focus and the directrix is \(4 - 2 = 2\).
- Hence, the vertex is 1 unit away from both the focus and the directrix. It lies on the line \(y = 3\) since that’s the midpoint between the focus and the directrix for the y-coordinate. Consequently, the vertex is at \((-3, 3)\).
3. **Equation of the Parabola:**
- Since the parabola opens vertically (upwards due to the focus being above the directrix), use the standard form for vertical parabolas:
\[
(x - h)^2 = 4p(y - k)
\]
where \((h, k)\) is the vertex.
- Here, \(h = -3\) and \(k = 3\).
- The distance \(p\) from the vertex to the focus is 1 (since 4 to 3 is 1 unit).
- Substitute to get the equation:
\[
(x + 3)^2 = 4(y - 3)
\]
This equation can be used to understand the parabola's position and shape with respect to the given focus and directrix.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda14e668-9ddd-4e75-9b43-53334816616b%2Fb7675a79-199e-4889-a0b4-2f1f7855275f%2Fb2munl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem:**
Find the standard form of the equation of the parabola with a focus at (-3, 4) and directrix \( y = 2 \).
**Explanation:**
The problem includes an empty coordinate grid with axes labeled.
To solve the problem, consider the following:
1. **Key points:**
- Focus: \((-3, 4)\)
- Directrix: \(y = 2\)
2. **Determine the Vertex:**
- The vertex of a parabola is equidistant from the focus and the directrix. The distance between the focus and the directrix is \(4 - 2 = 2\).
- Hence, the vertex is 1 unit away from both the focus and the directrix. It lies on the line \(y = 3\) since that’s the midpoint between the focus and the directrix for the y-coordinate. Consequently, the vertex is at \((-3, 3)\).
3. **Equation of the Parabola:**
- Since the parabola opens vertically (upwards due to the focus being above the directrix), use the standard form for vertical parabolas:
\[
(x - h)^2 = 4p(y - k)
\]
where \((h, k)\) is the vertex.
- Here, \(h = -3\) and \(k = 3\).
- The distance \(p\) from the vertex to the focus is 1 (since 4 to 3 is 1 unit).
- Substitute to get the equation:
\[
(x + 3)^2 = 4(y - 3)
\]
This equation can be used to understand the parabola's position and shape with respect to the given focus and directrix.
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