Question

Prove that the vertex is the point on a parabola closest to the focus.

Short answer

The vertex is the closest point to the focus due to symmetry and definition of a parabola.

Step-by-step solution

Understanding the Parabola

A parabola is defined as the set of all points equidistant from a focus point and a directrix line. The standard form of a parabola with the vertex at the origin is given by \( y = ax^2 \) (assuming the parabola opens upwards). The focus for this parabola is at the point \( (0, \frac{1}{4a}) \).

Vertex and Focus Distance

The vertex of the parabola, given by the point \((0,0)\), is directly below or above the focus. The distance from the vertex to the focus of the parabola \( y = ax^2 \) is \( \frac{1}{4a} \).

Analyzing Other Points

Consider another point on the parabola \((x, ax^2)\). The distance from this point to the focus \((0, \frac{1}{4a})\) is given by the distance formula: \[ d = \sqrt{x^2 + \left(ax^2 - \frac{1}{4a}\right)^2} \].

Minimizing the Distance

To find the point closest to the focus, differentiate the distance \(d\) with respect to \(x\) and find where the derivative equals zero. However, it's already known that at \(x=0\), the distance \(d\) is a minimum due to symmetry and the definition of a parabola.

Key concepts

Vertex

The vertex of a parabola is a crucial point that acts as its peak or dip depending on how the parabola opens. For a vertical parabola in the form of \( y = ax^2 \), the vertex is situated at \((0,0)\) if the parabola opens upwards or downwards. It represents the point of symmetry and is considered the turning point of the parabola.

The vertex is significant beyond just its position in the graph. It is a central reference point in many calculations, such as finding the minimum or maximum value of the quadratic function that defines the parabola. This makes the vertex a central concept in understanding parabolic shapes.

Focus

The focus of a parabola is a fixed point that, along with the directrix, defines the parabola. This point plays a key role in the geometric definition of the parabola: any point on the parabola is equidistant to the focus and the directrix.

For the parabola \( y = ax^2 \) with its vertex at the origin, the focus is positioned at \((0, \frac{1}{4a})\). This location just above (or below) the vertex is crucial in deriving properties of the parabola and understanding its geometric behavior.

Distance Formula

The distance formula is fundamental in many areas of geometry, helping to calculate the distance between any two points in a plane. It is derived from the Pythagorean theorem and is expressed as:

• \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

In the context of the parabola problem, the distance formula is used to find the distance between any point on the parabola and the focus.

For example, consider a point \((x, ax^2)\) on the parabola and the focus \((0, \frac{1}{4a})\). The distance between them is determined by substituting these coordinates into the distance formula, resulting in:

• \( d = \sqrt{x^2 + \left(ax^2 - \frac{1}{4a}\right)^2} \)

Understanding this formula is key to solving problems related to finding the closest points on a parabola to its focus.

Differentiation

Differentiation is a calculus technique used to find instantaneous rates of change or to determine the slopes of curves at any given point. In this scenario, differentiation aids in minimizing the distance between a point on the parabola and the focus.

To find this minimum, the derivative of the distance \(d(x)\) with respect to \(x\) is calculated. This process helps identify critical points where the derivative equals zero, which indicates potential minima or maxima. However, the symmetry of the parabola assures that at \(x=0\), this derivative results in a minimum distance.

Differentiation informs us not only where the minimum occurs but also enhances our understanding of the behavior of functions represented by parabolic equations.