Problem 77 - Precalculus

Question

Determine whether the statement is true or false. Justify your answer. It is possible for a parabola to intersect its directrix.

Step-by-Step Solution

Answer

False, a parabola cannot intersect its directrix. Because by definition, all points on a parabola are equidistant to its focus and directrix and if it intersects, it contradicts the definition.

Step 1: Understand the properties of a parabola

A parabola is defined as the set of all points that are equidistant to a fixed point (the focus) and a fixed line (the directrix). This means that for any point on the parabola, its distance to the focus is equal to its distance to the directrix.

Step 2: Analyze Intersection of a Parabola and its Directrix

If a parabola were to intersect its own directrix, there would exist a point on the parabola that is also on the directrix. But for this point, its distance to the focus and the directrix cannot be equal because it lies on the directrix, and therefore its distance to the directrix is 0. This contradicts the definition of a parabola.

Step 3: Conclude from analysis

Because the properties of a parabola contradict the possibility of a parabola intersecting its directrix, we can conclude that a parabola cannot intersect its directrix.

Key Concepts

Directrix

The directrix is a critical component in the definition of a parabola. It is a fixed line that helps to shape the parabola alongside the focus. To understand how it works, consider this: every point on a parabola is equidistant from the directrix and a point called the focus.
The directrix is unique in that it guides the curvature of the parabola without ever being intersected by it. This is because no point on the parabola can simultaneously lie on the parabola and be equidistant to both the directrix and the focus.
Some important properties of the directrix include:

• The directrix is always perpendicular to the axis of symmetry of the parabola.
• It does not touch or cross the parabola, ensuring its distance rules are not violated.
• Its position relative to the vertex helps in determining the position of the focus.

Focus

The focus of a parabola is a fixed point that, together with the directrix, defines the actually bent form that is the parabola. Each point on a parabola maintains the property of being equidistant from the focus and the directrix.
This unique point ensures that the parabola has a consistent shape — opening upwards, downwards, or sideways, depending on its orientation.
Here are a few handy facts about the focus of a parabola:

• The distance between the focus and the vertex is called the focal length.
• The focus lies along the parabola's axis of symmetry.
• Its unique relation with the directrix ensures that the parabola never intersects the directrix.

Parabola Properties

Parabolas are fascinating curves with distinct properties dictated by their focus and directrix. These properties help us understand why parabolas behave the way they do.
One of the essential properties is that every point on the curve is equidistant from both the focus and the directrix. This fundamental rule helps establish the parabolic shape.
Additional properties of parabolas include:

• Axis of symmetry: A parabola has a vertical or horizontal symmetry line, depending on its orientation, passing through the vertex and the focus.
• Vertex: This is the point where the parabola is closest to or furthest from the directrix, acting as the "turning point."
• Standard equation: For a parabola opening upwards or downwards, the formula is typically written as \(y = ax^2 + bx + c\).

These properties explain why a parabola cannot intersect its directrix, as doing so would break the equidistance rule.