Question
Identify the focus and directrix of the parabola represented by the equation (y-2)^2= -4 (x+2)
Answers
GPT 3.5
The given equation of the parabola is in the form (y - k)^2 = 4p(x - h), where (h, k) is the vertex and p is the distance between the vertex and focus/directrix.
Comparing the equation (y - 2)^2 = -4(x + 2) with the standard form, we have:
(h, k) = (-2, 2)
4p = -4
Since 4p is negative, the parabola opens to the left. The vertex is (-2, 2).
To find the focus and directrix, we need to determine the value of p.
Dividing both sides of the equation -4p = -4 by -4, we get:
p = 1
Since p = 1, the distance between the vertex and focus/directrix is 1 unit.
For a parabola that opens to the left, the focus is located to the left of the vertex and the directrix is a vertical line to the right of the vertex.
Therefore, the focus is (-3, 2), and the directrix is the vertical line x = -1.
Thus, the focus of the parabola represented by the equation (y - 2)^2 = -4(x + 2) is (-3, 2), and the directrix is x = -1.
Comparing the equation (y - 2)^2 = -4(x + 2) with the standard form, we have:
(h, k) = (-2, 2)
4p = -4
Since 4p is negative, the parabola opens to the left. The vertex is (-2, 2).
To find the focus and directrix, we need to determine the value of p.
Dividing both sides of the equation -4p = -4 by -4, we get:
p = 1
Since p = 1, the distance between the vertex and focus/directrix is 1 unit.
For a parabola that opens to the left, the focus is located to the left of the vertex and the directrix is a vertical line to the right of the vertex.
Therefore, the focus is (-3, 2), and the directrix is the vertical line x = -1.
Thus, the focus of the parabola represented by the equation (y - 2)^2 = -4(x + 2) is (-3, 2), and the directrix is x = -1.
oobleck
sorry. If 4p = -4, then p = -1
so the focus is at (-3,2)
and the directrix is x = -1
so the focus is at (-3,2)
and the directrix is x = -1
GPT 3.5
Apologies for the mistake in the calculation. You are correct.
Since 4p = -4, dividing both sides by 4 yields p = -1. Therefore, the distance between the vertex and focus/directrix is 1 unit.
The focus is located to the left of the vertex, so it is at (-3, 2). The directrix is a vertical line to the right of the vertex, which is x = -1.
Thus, the correct focus of the parabola represented by the equation (y - 2)^2 = -4(x + 2) is indeed (-3, 2), and the directrix is x = -1.
Since 4p = -4, dividing both sides by 4 yields p = -1. Therefore, the distance between the vertex and focus/directrix is 1 unit.
The focus is located to the left of the vertex, so it is at (-3, 2). The directrix is a vertical line to the right of the vertex, which is x = -1.
Thus, the correct focus of the parabola represented by the equation (y - 2)^2 = -4(x + 2) is indeed (-3, 2), and the directrix is x = -1.