What is the directrix and focus of the equation 1/16(y+4)^2=x-3
To find the directrix and focus of the given equation, which is in the standard form of a parabola, you can follow these steps:
Step 1: Identify the vertex of the parabola
The equation of the parabola is given as 1/16(y+4)^2 = x - 3. Notice that the term involving y is squared and the term involving x has a coefficient of 1, indicating that the parabola opens towards the right. The vertex of the parabola can be identified by shifting the (h, k) values, where h corresponds to x and k corresponds to y.
In this case, the equation is written in the form (y-k)^2 = 4p(x-h), where (h, k) represents the vertex. Comparing this with the given equation, we can determine that the vertex is (3, -4).
Step 2: Determine the distance between the vertex and the focus/directrix
For a parabola, the distance between the vertex and the focus (p) is equal to the distance between the vertex and the directrix.
In the given equation, the coefficient of x, which is 1/16, determines the value of p. Since 4p represents the distance between the vertex and the focus/directrix, we can deduce that p = 1/(4*1/16) = 4.
Step 3: Find the focus
The focus of the parabola can be found by adding the value of p to the x-coordinate of the vertex (h).
In this case, the x-coordinate of the vertex is 3, and p is 4. Therefore, the x-coordinate of the focus is 3 + 4 = 7. Thus, the focus of the parabola is (7, -4).
Step 4: Determine the directrix
The directrix is a line parallel to the y-axis and is at a distance p from the vertex. In this case, since the parabola opens towards the right, the directrix will be a vertical line located to the left of the vertex.
The equation of the directrix can be found by subtracting p from the x-coordinate of the vertex (h).
In this case, the x-coordinate of the vertex is 3, and p is 4. Therefore, the x-coordinate of the directrix is 3 - 4 = -1. Hence, the directrix can be represented as x = -1.
To summarize, the focus of the parabola is (7, -4), and the equation of the directrix is x = -1.