Question
A parabola, with its vertex at the origin, has a directrix at y = 3.
Which statements about the parabola are true? Select two options.
The focus is located at (0,–3).
The parabola opens to the left.
The p value can be determined by computing 4(3).
The parabola can be represented by the equation x2 = –12y.
The parabola can be represented by the equation y2 = 12x.
Which statements about the parabola are true? Select two options.
The focus is located at (0,–3).
The parabola opens to the left.
The p value can be determined by computing 4(3).
The parabola can be represented by the equation x2 = –12y.
The parabola can be represented by the equation y2 = 12x.
Answers
GPT-4o mini
To analyze the given parabola with its vertex at the origin (0,0) and a directrix at \( y = 3 \), we need to determine several characteristics about the orientation and equations related to the parabola.
1. **Focus and Directrix Relationship**: For a parabola, the focus and directrix are positioned such that the distance from any point on the parabola to the focus is equal to its distance to the directrix. Since the directrix is at \( y = 3 \) and the vertex is at the origin, the focus must be located downward at equal distance from the vertex. This gives a distance \( p \) equal to 3. Therefore, the focus is located at (0, -3).
2. **Opening Direction**: Since the directrix is above the vertex (at \( y = 3 \)), the parabola opens in the downward direction (towards negative \( y \)).
3. **Value of \( p \)**: The value of \( p \) is indeed 3 (the distance from the vertex to the directrix). The relationship for a parabola involving the value of \( p \) is given by \( 4p \). Hence, \( 4(3) = 12 \) gives a scale factor related to the parabola's equation.
4. **Equation of the Parabola**: Given that the parabola opens downward, the standard form of the equation will be:
\[
x^2 = -4py
\]
Substituting \( p = 3 \), we have:
\[
x^2 = -12y
\]
From the analyses above, we can conclude the following true statements:
- The focus is located at (0, –3).
- The parabola can be represented by the equation \( x^2 = -12y \).
Based on this reasoning, the correct answers are:
1. **The focus is located at (0,–3).**
2. **The parabola can be represented by the equation \( x^2 = –12y \).**
1. **Focus and Directrix Relationship**: For a parabola, the focus and directrix are positioned such that the distance from any point on the parabola to the focus is equal to its distance to the directrix. Since the directrix is at \( y = 3 \) and the vertex is at the origin, the focus must be located downward at equal distance from the vertex. This gives a distance \( p \) equal to 3. Therefore, the focus is located at (0, -3).
2. **Opening Direction**: Since the directrix is above the vertex (at \( y = 3 \)), the parabola opens in the downward direction (towards negative \( y \)).
3. **Value of \( p \)**: The value of \( p \) is indeed 3 (the distance from the vertex to the directrix). The relationship for a parabola involving the value of \( p \) is given by \( 4p \). Hence, \( 4(3) = 12 \) gives a scale factor related to the parabola's equation.
4. **Equation of the Parabola**: Given that the parabola opens downward, the standard form of the equation will be:
\[
x^2 = -4py
\]
Substituting \( p = 3 \), we have:
\[
x^2 = -12y
\]
From the analyses above, we can conclude the following true statements:
- The focus is located at (0, –3).
- The parabola can be represented by the equation \( x^2 = -12y \).
Based on this reasoning, the correct answers are:
1. **The focus is located at (0,–3).**
2. **The parabola can be represented by the equation \( x^2 = –12y \).**