Find the focus, directrix, and focal diameter of the parabola.
x = −4y2
I answered everything right except the directrix. which my ans was -1/16.
To find the focus, directrix, and focal diameter of the parabola defined by the equation x = -4y^2, we can convert it to the standard form of the equation of a parabola, which is (x - h) = 4p(y - k)^2.
Comparing the given equation x = -4y^2 to the standard form, we can see that the vertex of the parabola is at the origin (0, 0). Therefore, h = k = 0.
Now, to find the value of p, we consider the coefficient of y^2 in the equation. Notice that it is negative, indicating that the parabola opens to the left. In the standard form, p represents the distance between the vertex and the focus (in this case, the focus is on the left side).
Since the parabola opens to the left, p will be a positive value. We can rewrite the given equation as x = -4(y - 0)^2.
By comparing it with the standard form, we have (x - 0) = 4p(y - 0)^2. Thus, we can conclude that p = 1/4.
Now we have all the necessary information to find the focus, directrix, and focal diameter.
1. Focus:
The focus is a point on the axis of symmetry, which is the y-axis in this case.
Since the focus is p = 1/4 to the left of the vertex, the coordinates of the focus are (−1/4, 0).
2. Directrix:
The directrix is a vertical line that is p = 1/4 to the right of the vertex.
Since the vertex is at the origin, the equation of the directrix is x = 1/4.
3. Focal Diameter:
The focal diameter is the distance between the focus and a point on the parabola, passing through the focus and perpendicular to the axis of symmetry.
Since the focus is at (−1/4, 0) and the axis of symmetry is the y-axis, the focal diameter will be twice the distance from the focus to the x-axis.
The distance from the focus to the x-axis is the absolute value of the y-coordinate of the focus, which is 0. Therefore, the focal diameter is 2 * 0 = 0.
In summary:
The focus of the parabola x = -4y^2 is at (−1/4, 0).
The directrix is the line x = 1/4.
The focal diameter is 0.
Please note that there might be a mistake in your calculation for the directrix, which caused the incorrect answer. The correct directrix for this parabola is x = 1/4.