Find the vertex, focus and directrix of the parabola.
4x - y^2 - 2y - 33 = 0
y^2+2y+1 = -4x+33+1
(y+1)^2 = -4(x+8.5)
vertex:(-1,-8.5)
focus:(-2,-8.5)
Directrix: x=0
Is this correct?
y^2 + 2 y = 4 x - 33
y^2 + 2 y + 1 = 4 x - 32
(y+1)^2 = 4 (x-8)
(8,-1) is the vertex etc.
opens right, a = 4
vertex to focus = 4 so focus at (12,-1)
vertex to directrix = 4 so directrix at x = 4
No, your calculations are not correct.
To find the vertex, focus, and directrix of a parabola, we need to put the equation in the standard form:
y^2 = 4px
Given equation: 4x - y^2 - 2y - 33 = 0
Rearranging the equation, we get:
4x - y^2 - 2y = 33
Completing the square for the y terms, we add and subtract (2/2)^2 = 1 to the equation:
4x - (y^2 + 2y + 1) = 33 + 1
Simplifying, we have:
4x - (y+1)^2 = 34
Dividing throughout by 4, we get the equation in the desired standard form:
x - (1/4)(y+1)^2 = 8.5
Comparing this with the standard equation y^2 = 4px, we can see that p = -1/4.
The vertex of the parabola is given by the coordinates (h, k), where h = 0 and k = 8.5. Therefore, the vertex is (0, 8.5).
The focus of the parabola is at a distance of p = -1/4 from the vertex. Since the vertex is at (0, 8.5), the focus is at (0, 8.5 - 1/4) = (0, 8.25).
The directrix is a horizontal line located at a distance of p = -1/4 from the vertex. Since the vertex is at (0, 8.5), the equation of the directrix is x = -1/4.
Therefore, the correct answers are:
Vertex: (0, 8.5)
Focus: (0, 8.25)
Directrix: x = -1/4
To find the vertex, focus, and directrix of a parabola, we can start by rewriting the equation of the parabola in the standard form.
The given equation is 4x - y^2 - 2y - 33 = 0. To rewrite it in standard form, we need to complete the square.
Starting with the given equation, let's rearrange the terms:
4x - y^2 - 2y - 33 = 0
4x - (y^2 + 2y) - 33 = 0
Next, we want to complete the square for the y terms. To do this, we need to add and subtract a constant inside the parentheses. The constant we need to add and subtract can be found by taking half of the coefficient of y (in this case, it is 2) and squaring it (2^2 = 4).
4x - (y^2 + 2y + 1 - 1) - 33 = 0
4x - ((y + 1)^2 - 1) - 33 = 0
4x - (y + 1)^2 + 1 - 33 = 0
4x - (y + 1)^2 - 32 = 0
Now we have the equation in vertex form: 4x - (y + 1)^2 = 32
Comparing this equation to the standard form of a parabola, which is (x - h) = 4p(y - k)^2, we can see that the vertex is (-h, k) and the focus is (-h, k + p), where p is the distance between the vertex and the focus, also known as the focal length. The directrix is a line parallel to the y-axis and is given by the equation x = h - p.
From our equation 4x - (y + 1)^2 = 32, we can determine that h = 0 and k = -1.
So, the vertex is (-h, k) = (0, -1).
The focal length (p) can be found by dividing the coefficient of x (which is 4) by 4p. In this case, 4p = 32, which means p = 8.
The focus is then given by (-h, k + p) = (0, -1 + 8), which simplifies to (0, 7).
The directrix is a vertical line parallel to the y-axis and is given by x = h - p. In this case, x = 0 - 8, so the directrix is x = -8.
Therefore, the vertex is (0, -1), the focus is (0, 7), and the directrix is x = -8.
So, based on the calculations, your answers are incorrect. The correct answers are:
Vertex: (0, -1)
Focus: (0, 7)
Directrix: x = -8