Chapter11: Conics
Section11.2: Parabolas
Problem 97E: Write the equation of a parabola that opens up or down in standard form and the equation of a...
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![### Problem Statement
**Question:**
Find an equation of the parabola with vertex (2, 2) and directrix \( x = 8 \).
### Solution Approach
The general form of a parabola with a vertical axis or horizontal axis can be derived based on its vertex and the directrix.
In this case, the vertex is given as \((2, 2)\), and the directrix is given as \( x = 8 \).
1. **Understanding the Parabola Orientation**:
- Since the directrix \( x = 8 \) is a vertical line, this implies that the parabola opens towards the left or right along the x-axis.
2. **Equation of a Parabola Opening Horizontally**:
- The standard form for a parabola opening horizontally is \((y - k)^2 = 4p (x - h)\), where \((h, k)\) is the vertex, and \(p\) is the distance from the vertex to the directrix (or focus).
3. **Calculating \( p \)**:
- The distance \( p \) from the vertex \((2, 2)\) to the directrix \( x = 8 \) is:
\[ p = Distance = |8 - 2| = 6 \]
4. **Determining the Direction**:
- If the parabola opens to the left, the equation will include a negative value for \( p \).
- Given that \( x = 8 \) is to the right of \( x = 2 \), the parabola will open leftwards (negative \( p \)).
5. **Substituting the Values**:
- Here, the vertex \((h, k) = (2, 2)\) and \( p = -6 \) (since it opens to the left).
\[ (y - 2)^2 = 4(-6)(x - 2) \]
\[ (y - 2)^2 = -24(x - 2) \]
Therefore, the equation of the parabola is:
\[ (y - 2)^2 = -24(x - 2) \]
### Interactive Elements:
The interface includes options like:
- Square
- Horizontal Line
- Vertical Line
- Deleting entries (X)
- Resetting inputs (arrow)
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Transcribed Image Text:### Problem Statement
**Question:**
Find an equation of the parabola with vertex (2, 2) and directrix \( x = 8 \).
### Solution Approach
The general form of a parabola with a vertical axis or horizontal axis can be derived based on its vertex and the directrix.
In this case, the vertex is given as \((2, 2)\), and the directrix is given as \( x = 8 \).
1. **Understanding the Parabola Orientation**:
- Since the directrix \( x = 8 \) is a vertical line, this implies that the parabola opens towards the left or right along the x-axis.
2. **Equation of a Parabola Opening Horizontally**:
- The standard form for a parabola opening horizontally is \((y - k)^2 = 4p (x - h)\), where \((h, k)\) is the vertex, and \(p\) is the distance from the vertex to the directrix (or focus).
3. **Calculating \( p \)**:
- The distance \( p \) from the vertex \((2, 2)\) to the directrix \( x = 8 \) is:
\[ p = Distance = |8 - 2| = 6 \]
4. **Determining the Direction**:
- If the parabola opens to the left, the equation will include a negative value for \( p \).
- Given that \( x = 8 \) is to the right of \( x = 2 \), the parabola will open leftwards (negative \( p \)).
5. **Substituting the Values**:
- Here, the vertex \((h, k) = (2, 2)\) and \( p = -6 \) (since it opens to the left).
\[ (y - 2)^2 = 4(-6)(x - 2) \]
\[ (y - 2)^2 = -24(x - 2) \]
Therefore, the equation of the parabola is:
\[ (y - 2)^2 = -24(x - 2) \]
### Interactive Elements:
The interface includes options like:
- Square
- Horizontal Line
- Vertical Line
- Deleting entries (X)
- Resetting inputs (arrow)
- Help/
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