write y2-14x-4y-66=0 in standard form. Identify the focus, vertex, axsis of symmetry, and directrix. (This one has me completely stumped good luck)
To rewrite the equation in standard form, we'll need to complete the square for both the x and y terms.
Let's begin by rearranging the equation:
y^2 - 4y - 14x - 66 = 0
Now, let's group the x and y terms:
(y^2 - 4y) - 14x - 66 = 0
To complete the square for the y terms, we'll need to add and subtract the square of half the coefficient of y. In this case, the coefficient is -4, so we'll add and subtract (4/2)^2 = 4:
(y^2 - 4y + 4) - 4 - 14x - 66 = 0
Next, we can combine the constants:
(y^2 - 4y + 4) - 14x - 70 = 0
Now, let's simplify the y-term:
(y - 2)^2 - 14x - 70 = 0
To complete the square for the x term, we'll need to add and subtract the square of half the coefficient of x. In this case, the coefficient is -14, so we'll add and subtract (14/2)^2 = 49:
(y - 2)^2 - 14x + 49 - 49 - 70 = 0
Simplifying further:
(y - 2)^2 - 14x + 49 - 119 = 0
(y - 2)^2 - 14x - 70 = 0
Now the equation is in standard form: (y - h)^2 = 4p(x - k)
Comparing this to the given equation, we can determine the values of h, k, and p:
- h = 2
- k = -70 / (-14) = 5
- 4p = -14
From 4p = -14, we find that p = -14 / 4 = -7/2
Therefore, the equation in standard form is (y - 2)^2 = -14(x - 5).
Now, let's identify the elements of the parabolic equation:
1. Vertex: The vertex of the parabola is given by the coordinates (h, k). So, the vertex is V(5, 2).
2. Axis of Symmetry: The axis of symmetry is a vertical line passing through the vertex. So, the equation of the axis of symmetry is x = 5.
3. Focus: The focus of the parabola lies on the axis of symmetry and is equidistant from the vertex and the directrix. Since the parabola is opening to the left, the focus will be p units to the left of the vertex. So, the focus is F(5 - 7/2, 2) = F(-3/2, 2).
4. Directrix: The directrix is a vertical line that is perpendicular to the axis of symmetry and is equidistant from the vertex. Therefore, the directrix is x = 5 + 7/2 = 17/2.
In summary:
- Vertex: V(5, 2)
- Axis of Symmetry: x = 5
- Focus: F(-3/2, 2)
- Directrix: x = 17/2