Question
What is the vertex of a parabola that has a focus at (2,1)
and the directrix at y=−3
?(1 point)
vertex: (
,
)
Remaining Atte
and the directrix at y=−3
?(1 point)
vertex: (
,
)
Remaining Atte
Answers
Answered by
GPT-4o mini
To find the vertex of a parabola given the focus and the directrix, we need to determine the point halfway between the focus and the directrix.
The focus is at the point \( (2, 1) \), and the directrix is the line \( y = -3 \).
1. The y-coordinate of the focus is \( 1 \) and the y-coordinate of the directrix is \( -3 \).
2. The vertex lies halfway between these two y-coordinates, so we can calculate the y-coordinate of the vertex by averaging the two:
\[
y_{\text{vertex}} = \frac{y_{\text{focus}} + y_{\text{directrix}}}{2} = \frac{1 + (-3)}{2} = \frac{1 - 3}{2} = \frac{-2}{2} = -1
\]
3. The x-coordinate of the vertex is the same as the x-coordinate of the focus, since the parabola opens either upwards or downwards (in this case, it opens upwards since the focus is above the directrix).
Thus, the x-coordinate of the vertex is \( 2 \).
Putting it all together, the vertex of the parabola is:
\[
\text{vertex: } (2, -1)
\]
The focus is at the point \( (2, 1) \), and the directrix is the line \( y = -3 \).
1. The y-coordinate of the focus is \( 1 \) and the y-coordinate of the directrix is \( -3 \).
2. The vertex lies halfway between these two y-coordinates, so we can calculate the y-coordinate of the vertex by averaging the two:
\[
y_{\text{vertex}} = \frac{y_{\text{focus}} + y_{\text{directrix}}}{2} = \frac{1 + (-3)}{2} = \frac{1 - 3}{2} = \frac{-2}{2} = -1
\]
3. The x-coordinate of the vertex is the same as the x-coordinate of the focus, since the parabola opens either upwards or downwards (in this case, it opens upwards since the focus is above the directrix).
Thus, the x-coordinate of the vertex is \( 2 \).
Putting it all together, the vertex of the parabola is:
\[
\text{vertex: } (2, -1)
\]
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