Question

The mirror in an automobile headlight has a parabolic cross section, with the lightbulb at the focus. On a schematic, the equation of the parabola is given as \(x^{2}=4 y .\) At what coordinates should you place the lightbulb?

Short answer

The lightbulb should be placed at (0, 1).

Step-by-step solution

Understanding the equation

The equation of the parabola is given as \( x^2 = 4y \). This is in the form \( x^2 = 4py \), where \( p \) is the focal length. This tells us the parabola opens upwards with its vertex at the origin \((0,0)\).

Identifying the focal length

In the equation \( x^2 = 4py \), we compare it with the given equation \( x^2 = 4y \). Here, \( 4p = 4 \), so we solve for \( p \): \( p = \frac{4}{4} = 1 \).

Finding the focus coordinates

The focus of a parabola given by \( x^2 = 4py \) is \((0,p)\). Since we found \( p = 1 \) in the previous step, the focus, and therefore the place where the lightbulb should be, is at \((0, 1)\).

Key concepts

Focus of Parabola

The focus of a parabola is an essential point that determines the reflective properties of a parabolic mirror. A parabola is defined as the set of all points equidistant from a focus and a directrix line. This unique property makes parabolas ideal for focusing light or sound waves, which is why they are often used in telescope mirrors and automobile headlights. In the context of headlights, the light emitted from the bulb (placed at the focus) reflects off the parabolic surface in parallel rays, providing a concentrated and directed beam of light.
For a vertically oriented parabola, the focus lies along its axis of symmetry. For the given parabola with equation \(x^2 = 4y\), the focus is at \((0, p)\), where \(p\) is the distance from the vertex to the focus.
Understanding the focus helps us determine not just where to place a bulb in a headlight but also how the light will be projected, ensuring a bright and focused beam to illuminate the road ahead.

Parabola Equation

The equation of a parabola is critical in describing its geometric properties. The standard form of a parabola that opens vertically is given by \(x^2 = 4py\). Here, the coefficient \(4p\) relates directly to the parabola's width and position of the focus.
The equation provided, \(x^2 = 4y\), indicates a simple case where the parabola's vertex is at the origin \((0, 0)\) and opens upwards.

• The vertex, being the 'tip' of the parabola, serves as the point from which the curvature begins.
• The coefficient \(4\), in this instance, determines the parabolic spread and directly affects where the focus is located along the y-axis.

To solve such equations, comparing the format with \(x^2 = 4py\) helps identify \(p\), which is crucial for determining the focal length and ultimately the focus point.

Focal Length

The focal length of a parabola, often denoted \(p\), is the distance from the vertex to the focus. It is an important parameter that influences the parabola's ability to concentrate light or other forms of energy.
In the equation \(x^2 = 4py\), \(p\) is derived by evaluating \(4p\) against the coefficient of \(y\) in the equation.
For the equation \(x^2 = 4y\), our task was compute \(p\) as follows:

• Set \(4p = 4\), hence solving yields \(p = 1\).

This tells us the focus is 1 unit above the vertex at the point \((0, 1)\).
Recognizing the focal length is necessary to understand how the parabolic mirror will direct or focus light. In practical terms, placing a light source at this focal distance ensures the impact of a bright, evenly spread beam, enhancing visibility at night.