Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. The point on a parabola closest to the focus is the vertex.

Verified step by step guidance

1. Recall the definition of a parabola: it is the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix.
2. Understand that the vertex of the parabola is the point on the parabola that lies exactly halfway between the focus and the directrix.
3. To determine if the vertex is the closest point on the parabola to the focus, consider the distance from any point on the parabola to the focus and compare it to the distance from the vertex to the focus.
4. Use the geometric property that the distance from the vertex to the focus is the shortest distance from the parabola to the focus because the vertex lies on the axis of symmetry and minimizes this distance.
5. Conclude that the vertex is indeed the point on the parabola closest to the focus, since any other point on the parabola will be farther away due to the parabola's shape and definition.

Key Concepts

Definition of a Parabola and Its Focus

A parabola is the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix. The vertex is the point on the parabola closest to the directrix and lies midway between the focus and directrix.

Distance from a Point to the Focus

The distance from any point on the parabola to the focus varies along the curve. Understanding how this distance changes is essential to determine which point on the parabola is closest to the focus.

Optimization and Minimizing Distance

To find the point on the parabola closest to the focus, one must analyze the distance function and use calculus techniques like differentiation to find its minimum value, confirming whether the vertex is indeed the closest point.