Problem 91 - College Algebra with Modeling and Visualization

Question

A comet sometimes travels along a parabolic path as it passes the sun. In this case the sun is located at the focus of the parabola and the comet passes the sun once, rather than orbiting the sun. Suppose the path of a comet is given by \(y^{2}=100 x,\) where units are in millions of miles. (a) Find the coordinates of the sun. (b) Find the minimum distance between the sun and the comet.

Step-by-Step Solution

Answer

(a) The coordinates of the sun are (25, 0). (b) The minimum distance is 25 million miles.

Step 1: Identify the Parabola Characteristics

The equation given is \( y^2 = 100x \). This is a standard form of a parabola with the vertex at the origin \((0,0)\) and opening to the right. Based on its form, \( y^2 = 4px \), where \( p \) is the distance from the vertex to the focus.

Step 2: Calculate the Distance to Focus

For the equation \( y^2 = 100x \), compare it to \( y^2 = 4px \) to find \( 4p = 100 \). Solving \( 4p = 100 \) gives \( p = 25 \). This means the focus, where the sun is located, is at \( (25, 0) \).

Step 3: Identify the Path of the Comet

Since the parabola equation is centered at the origin \((0,0)\) and opens to the right, the comet travels from one side of the vertex at \(x = 0\) to the other side along the path determined by the equation \(y^2 = 100x\).

Step 4: Understand the Minimum Distance

The minimum distance between the focus and a point on the parabola is the distance from the focus to the vertex of the parabola. This is because the directrix doesn't affect distance in terms of the closest point but rather ensures the definition of the parabola.

Step 5: Employ Distance Formula

The distance from the vertex \((0,0)\) to the focus \((25,0)\) is the minimum distance. Since the points lie on the same x-axis, the distance is simply \( |25 - 0| = 25 \) million miles.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. Depending on the angle and position of the intersecting plane, these sections can form different shapes such as circles, ellipses, parabolas, and hyperbolas. These shapes have interesting and unique properties, making them essential for various applications in science and geometry.
Parabolas are one of the most common conic sections. They are known for their distinctive U-shaped curve that can open upwards, downwards, left, or right. A key feature of parabolas is their symmetry, often possessing a single axis of symmetry.
Each conic section is characterized by a specific standard equation. For parabolas, when opening left or right, it often appears as \( y^2 = 4px \), where \( p \) represents the distance from the vertex to the focus. The position and orientation of parabolas depend on the signs and parameters in their equations.

• Parabolas open left or right with equations of the form \( y^2 = 4px \).
• Parabolas open up or down with equations like \( x^2 = 4py \).
• Ellipses have a standard equation of \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).
• Hyperbolas feature the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \).

Understanding conic sections is pivotal for solving many geometric problems and finding paths, like that of a comet around the sun.

Focus of a Parabola

The focus is a special point associated with parabolas, playing a critical role in their geometry and application. It is used to describe how parabolas gather or spread light and establish orbits.
In mathematical terms, the focus of a parabola is a fixed point such that any point on the parabola is equidistant to the focus and a line called the directrix. This definition helps establish the criterion for the curve and makes understanding parabolas easier.
For the equation \( y^2 = 4px \), the focus is located at \( (p, 0) \) when the vertex is at the origin and the parabola opens to the right. Thus, knowing the structure of the parabola's equation helps quickly identify the location of this significant point.

• The focus helps in determining the position and shape of the parabola.
• It is integral to applications involving reflections and optics.
• In the exercise, the parabola's vertex is at the origin, and its focus is calculated at \( (25, 0) \) from the given equation \( y^2 = 100x \).

Understanding the role of the focus is essential for effectively using parabolas in practical and theoretical contexts.

Distance Calculation

Distance calculation in geometric problems often involves finding the shortest or specific distances between points, like from a point to a line or between two points.
The distance formula, \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \), enables the computation of straight-line distances in a plane. This is particularly helpful when working with coordinates and needing exact measurements.
In the context of our exercise, calculating the minimum distance between the sun (focus of the parabola) and the path of the comet involves understanding geometric relationships. For a right-opening parabola like \( y^2 = 100x \), the shortest distance is always from the vertex to the focus.

• To find the minimum distance, compute the formula for the given points: \( (0,0) \) and \( (25,0) \).
• Since these points lie on the same axis, the distance simplifies to \( |25| = 25 \) million miles.
• This calculation showcases understanding geometry principles and practical application of the distance formula.

Mastering distance calculations is fundamental in geometry and reinforces learning about shapes, paths, and spatial relationships.