Question
Identify the focus and directrix of the parabola represented by the equation left-parenthesis y minus 2 right-parenthesis squared equals negative 4 left-parenthesis x plus 2 right-parenthesis.
(1 point)
Responses
The focus is left-parenthesis negative 3 comma 2 right-parenthesis and the directrix is x equals negative 1.
The focus is Image with alt text: left-parenthesis negative 3 comma 2 right-parenthesis and the directrix is Image with alt text: x equals negative 1 .
The focus is left-parenthesis negative 2 comma 2 right-parenthesis and the directrix is x equals negative 1.
The focus is Image with alt text: left-parenthesis negative 2 comma 2 right-parenthesis and the directrix is Image with alt text: x equals negative 1 .
The focus is left-parenthesis 2 comma negative 2 right-parenthesis and the directrix is y equals negative 1.
The focus is Image with alt text: left-parenthesis 2 comma negative 2 right-parenthesis and the directrix is Image with alt text: y equals negative 1 .
The focus is left-parenthesis 2 comma negative 3 right-parenthesis and the directrix is y equals negative 1.
(1 point)
Responses
The focus is left-parenthesis negative 3 comma 2 right-parenthesis and the directrix is x equals negative 1.
The focus is Image with alt text: left-parenthesis negative 3 comma 2 right-parenthesis and the directrix is Image with alt text: x equals negative 1 .
The focus is left-parenthesis negative 2 comma 2 right-parenthesis and the directrix is x equals negative 1.
The focus is Image with alt text: left-parenthesis negative 2 comma 2 right-parenthesis and the directrix is Image with alt text: x equals negative 1 .
The focus is left-parenthesis 2 comma negative 2 right-parenthesis and the directrix is y equals negative 1.
The focus is Image with alt text: left-parenthesis 2 comma negative 2 right-parenthesis and the directrix is Image with alt text: y equals negative 1 .
The focus is left-parenthesis 2 comma negative 3 right-parenthesis and the directrix is y equals negative 1.
Answers
GPT-4o mini
To identify the focus and directrix of the given parabola represented by the equation \((y - 2)^2 = -4(x + 2)\), we need to rewrite and analyze the equation.
The equation is in the form \((y - k)^2 = 4p(x - h)\), where \((h, k)\) is the vertex of the parabola and \(p\) determines the distance from the vertex to the focus and the directrix.
Here, we have:
- \(k = 2\)
- \(h = -2\)
- The equation has a negative sign, indicating that it opens to the left.
We can find \(p\) from \(-4\):
- \(4p = -4\) ⇒ \(p = -1\)
Thus, the vertex \((h, k)\) of the parabola is:
\[
(-2, 2)
\]
Since the parabola opens to the left, the focus, which is located \(p\) units left of the vertex, is at:
\[
(-2 - 1, 2) = (-3, 2)
\]
The directrix is a vertical line located \(p\) units to the right of the vertex, which is at:
\[
x = -2 + 1 = -1
\]
Now, we can summarize the results:
- Focus: \((-3, 2)\)
- Directrix: \(x = -1\)
Thus, the correct answer is:
**The focus is \((-3, 2)\) and the directrix is \(x = -1\).**
The equation is in the form \((y - k)^2 = 4p(x - h)\), where \((h, k)\) is the vertex of the parabola and \(p\) determines the distance from the vertex to the focus and the directrix.
Here, we have:
- \(k = 2\)
- \(h = -2\)
- The equation has a negative sign, indicating that it opens to the left.
We can find \(p\) from \(-4\):
- \(4p = -4\) ⇒ \(p = -1\)
Thus, the vertex \((h, k)\) of the parabola is:
\[
(-2, 2)
\]
Since the parabola opens to the left, the focus, which is located \(p\) units left of the vertex, is at:
\[
(-2 - 1, 2) = (-3, 2)
\]
The directrix is a vertical line located \(p\) units to the right of the vertex, which is at:
\[
x = -2 + 1 = -1
\]
Now, we can summarize the results:
- Focus: \((-3, 2)\)
- Directrix: \(x = -1\)
Thus, the correct answer is:
**The focus is \((-3, 2)\) and the directrix is \(x = -1\).**