Question

Find the vertex, focus, directrix, and focal width of the parabola:

y^2 = 4px

A) Vertex: (0, 0); Focus: (0, -3); Directrix: y = 3; Focal width: 48
B) Vertex: (0, 0); Focus: (-6, 0); Directrix: x = 3; Focal width: 48
C) Vertex: (0, 0); Focus: (0, 3); Directrix: y = -3; Focal width: 3
D) Vertex: (0, 0); Focus: (0, -3); Directrix: y = 3; Focal width: 12

Answer 1

The vertex of the parabola is at (0, 0), the directrix is x = 0, the focus is at (0, 0), and the focal width is 0.

Step-by-step explanation:

The given equation, y2 = 4px, represents a parabola. The general form of a parabola equation is y = ax2 + bx + c. Comparing it with the given equation, we can see that a = 0, b = 0, and c = 0.

Since a = 0, the parabola opens horizontally. The vertex of the parabola is at (0, 0) as both x and y terms are squared.

Since the coefficient of x is 0, the directrix is a vertical line passing through the focus. Therefore, the directrix is given by x = -p. In this case, p = 0, so the directrix is x = 0.

The focus of the parabola can be found using the formula (p, 0). Since p = 0, the focus is at (0, 0).

The focal width is the distance between the directrix and the focus. In this case, the focal width is 0 since the directrix and the focus coincide at x = 0.