x^2+4x+8y-4=0
Find the Vertex, Foci, Directrix, and Axis of Symmetry.
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To find the vertex, foci, directrix, and axis of symmetry of the equation x^2 + 4x + 8y - 4 = 0, we need to identify the type of conic section the equation represents. Let's rearrange the equation into the standard form for a conic section:
x^2 + 4x - 8y + 4 = 0
Now, we can complete the square to rewrite the equation in vertex form:
(x^2 + 4x) - 8y + 4 = 0
(x^2 + 4x + 4) - 8y = 0 + 4
(x + 2)^2 - 8y = 4
Next, let's determine the type of conic section. The equation is in the form (x - h)^2 = 4p(y - k), where (h, k) is the vertex of the parabola. Therefore, this equation represents a parabola.
From the equation, we can identify the vertex as (-2, k), where k is a constant term. In this case, k = 0, so the vertex is (-2, 0).
To find the axis of symmetry, we can simply take the x-coordinate of the vertex. Therefore, the axis of symmetry is x = -2.
To find the foci, we need to determine the value of p. In general, the distance from the vertex to the focus is equal to p. Comparing the equation to the standard form, we can see that 4p = -8, so p = -2.
In a parabola, the distance from the vertex to the focus is given by |p|. Therefore, the focal distance is 2 units. Since the parabola opens upward, the foci are located above the vertex. The foci will be at (-2, 2) and (-2, -2).
Lastly, to find the directrix, we need to determine a line that is |p| units away from the vertex and is parallel to the y-axis. Since p = -2, we can find the directrix by adding 2 to the y-coordinate of the vertex (-2, 0). Therefore, the directrix is the line y = 2.
To summarize:
- Vertex: (-2, 0)
- Foci: (-2, 2) and (-2, -2)
- Directrix: y = 2
- Axis of Symmetry: x = -2
To find the vertex, foci, directrix, and axis of symmetry for the equation x^2+4x+8y-4=0, we can first rewrite the equation in vertex form.
1. Convert to vertex form:
To convert the given equation into vertex form, we need to complete the square. Move the constant term (-4) to the other side of the equation:
x^2 + 4x + 8y = 4
Next, complete the square for the x-terms. Take half of the coefficient of x (which is 2) and square it (which gives us 4):
x^2 + 4x + 4 = 4 - 4
Add the square term (4) to both sides of the equation:
x^2 + 4x + 4 = -4 + 4
Factor the left side of the equation as a perfect square:
(x + 2)^2 = 0
2. Determine the vertex:
The vertex form of the equation is (x - h)^2 = 4p(y - k), where (h, k) represents the coordinates of the vertex. In this case, the equation is already in vertex form, so the vertex is (-2, 0).
3. Find the axis of symmetry:
The axis of symmetry is a vertical line that passes through the vertex. In this case, the equation of the axis of symmetry is x = -2.
4. Determine the value of p:
The value of p represents the distance between the vertex and the focus point (or between the vertex and the directrix). In this case, p is not explicitly given. However, we can find it by comparing the equation to the standard form (x - h)^2 = 4p(y - k).
Comparing the equation to the standard form, we can see that (x + 2)^2 = 4p(y - 0), which gives us p = 1/4.
5. Find the foci:
The foci are points on the parabola that lie along the axis of symmetry and are equidistant to the vertex. To find the foci, we need to know the value of p.
Using the value of p we determined earlier (p = 1/4), the coordinates of the foci are (-2, 1/4) and (-2, -1/4).
6. Find the directrix:
The directrix is a horizontal line that is perpendicular to the axis of symmetry and is equidistant from the vertex as the foci. In this case, the directrix is a line parallel to the x-axis.
Since the vertex is at (h, k) = (-2, 0) and the value of p is 1/4, the equation of the directrix is y = -1/4.
Summary:
- Vertex: (-2, 0)
- Axis of symmetry: x = -2
- Foci: (-2, 1/4) and (-2, -1/4)
- Directrix: y = -1/4