Question
what is the vertex of parabola that has a focus at (2,1) and the directrix at y = -3?
vertex: (__,__)
vertex: (__,__)
Answers
GPT-4o mini
To find the vertex of a parabola given its focus and directrix, we can use the property that the vertex is located midway 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 vertex is the average of the y-coordinate of the focus and the y-coordinate of the directrix:
\[
y = \frac{1 + (-3)}{2} = \frac{1 - 3}{2} = \frac{-2}{2} = -1.
\]
2. The x-coordinate of the vertex will be the same as the x-coordinate of the focus since the parabola opens vertically (it is oriented with respect to the y-axis):
\[
x = 2.
\]
Therefore, the vertex of the parabola is at the point \((2, -1)\).
Final answer: 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 vertex is the average of the y-coordinate of the focus and the y-coordinate of the directrix:
\[
y = \frac{1 + (-3)}{2} = \frac{1 - 3}{2} = \frac{-2}{2} = -1.
\]
2. The x-coordinate of the vertex will be the same as the x-coordinate of the focus since the parabola opens vertically (it is oriented with respect to the y-axis):
\[
x = 2.
\]
Therefore, the vertex of the parabola is at the point \((2, -1)\).
Final answer: vertex: \((2, -1)\)