Find the standard form of the equation of each parabola satisfying the given conditions. Focus: (3, 2); Directrix: x = - 1

Verified step by step guidance

1. Recall that the standard form of a parabola depends on its orientation. Since the directrix is a vertical line (x = -1), the parabola opens horizontally (either left or right).
2. The vertex of the parabola lies exactly halfway between the focus and the directrix. Calculate the x-coordinate of the vertex by finding the midpoint between the focus's x-coordinate (3) and the directrix's x-value (-1): \(\frac{3 + (-1)}{2}\).
3. The y-coordinate of the vertex is the same as the y-coordinate of the focus, which is 2, because the parabola opens horizontally.
4. Determine the distance \(p\) between the vertex and the focus (or vertex and directrix). This distance is the absolute difference between the x-coordinates of the vertex and the focus.
5. Use the standard form of a horizontally opening parabola: \[(y - k)^2 = 4p(x - h)\] where \((h, k)\) is the vertex. Substitute the vertex coordinates and the value of \(p\) (positive if the parabola opens to the right, negative if to the left) to write the equation.

Key Concepts

Definition of a Parabola

A parabola is the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix. Understanding this definition helps in deriving the equation of the parabola based on the given focus and directrix.

Standard Form of a Parabola

The standard form of a parabola's equation depends on its orientation. For a parabola with a vertical axis, the form is (x - h)^2 = 4p(y - k), and for a horizontal axis, it is (y - k)^2 = 4p(x - h), where (h, k) is the vertex and p is the distance from the vertex to the focus or directrix.

Finding the Vertex and Parameter p

The vertex lies midway between the focus and directrix. The distance p is the distance from the vertex to the focus (positive if the parabola opens towards the focus). Calculating these allows you to write the parabola's equation in standard form.