Question

Give the property that defines all parabolas.

Short answer

Question: Provide a brief definition of a parabola and explain the roles of the focus and directrix in its formation. Answer: A parabola is defined as the set of all points in a plane that are equidistant from a fixed point called the focus and a fixed straight line called the directrix. The focus is a point within the parabola's interior, while the directrix is a straight line outside the parabola. The parabola's shape is determined by the symmetrical relationship between all points on the curve and their equal distance from the focus and directrix.

Step-by-step solution

Property Definition

A parabola is defined as the set of all points in a plane that are equidistant from a fixed point (called the focus) and a fixed straight line (called the directrix).

Understanding the Focus

In a parabola, the focus is a fixed point that lies within the parabola's interior. The distance between any point on the parabola and the focus is called the focal length.

Understanding the Directrix

The directrix is a fixed straight line that lies outside the parabola. Every point on the parabola is equidistant to the focus and the directrix.

The Parabola Equation

For a parabola with a vertical axis of symmetry and focus at (h, k + p) and directrix y = k - p, we can write its equation as: \[ (x - h)^2 = 4p(y - k) \] To find the equation for a parabola with a horizontal axis of symmetry and focus at (h + p, k) and directrix x = h - p, we can use the equation: \[ (y - k)^2 = 4p(x - h) \]

Visualizing a Parabola

Imagine a graph with a focus at point F and a directrix as a straight line parallel to the x-axis or y-axis. The parabola's shape will be a curve where all points on the parabola mirror each other across the vertical or horizontal line of symmetry. The parabola is symmetric and ordinarily shaped like a U or an inverted U (for vertical parabolas), or shaped like an opening to the right or to the left for (horizontal parabolas).