what will be the focus and directrix of parabola y*y = -16x.
Given: y*y = -16x.
y^2 = -16x,
x = -(1/16)y^2, x-parabola.
a = -(1/16),
4a = -(4/16) = -(1/4),
1/4a = -4.
[1/4a] = 4.
V(h,k), k = Yv = -b/2a = 0/2a = 0.
h = -(1/16)0^2 = 0.
F(X1,0), V(0,0), D(X2,0).
X1 = k - [1/4a] = 0 - 4 = -4.
X2 = k + [1/4a] = 0 + 4 = 4.
F(-4,0), V(0,0), D(4,0).
To determine the focus and directrix of a parabola, we can use the standard form of the equation for a vertical parabola: (y - k)^2 = 4p(x - h), where (h, k) represents the vertex and p is the distance between the vertex and the focus (as well as the vertex and the directrix). Let's find the standard form equation for the given parabola y^2 = -16x.
First, rewrite the equation as (y - 0)^2 = -16(x - 0). This simplifies to y^2 = -16x.
From this equation, we can see that the vertex is located at the origin (0, 0), which is (h, k) in the standard form equation.
To determine the value of p, we need to find the distance between the vertex and the focus or the vertex and the directrix. Since the parabola opens to the left (the coefficient of x is negative), the focus will be to the right of the vertex. The distance between the vertex and the focus is p, while the distance between the vertex and the directrix is -p.
The formula for the distance between the vertex and the focus (or the directrix) for a parabola in standard form is |4p|.
In this case, |4p| = 16, so 4p = ±16.
Solving for p, we divide both sides by 4, resulting in p = ±4.
Since the parabola opens to the left, the focus will be p units to the right of the vertex. Therefore, the focus is located at (p, k), which is (4, 0).
Similarly, the directrix is p units to the left of the vertex. Therefore, the directrix is a vertical line at x = -p, which is x = -4.
In conclusion, for the parabola y^2 = -16x, the focus is located at the point (4, 0), and the directrix is the vertical line x = -4.