Problem 31 - Essential Calculus

Question

Eliminate the parameter and obtain the standard form of the rectangular equation. $$ \begin{aligned} &\text { Line through }\left(x_{1}, y_{1}\right) \text { and }\left(x_{2}, y_{2}\right)\\\ &x=x_{1}+t\left(x_{2}-x_{1}\right), \quad y=y_{1}+t\left(y_{2}-y_{1}\right) \end{aligned} $$

Step-by-Step Solution

Answer

The rectangular form of the given line is \( (x - x_{1}) (y_{2} - y_{1}) = (y - y_{1}) (x_{2} - x_{1}) \)

Step 1: Solve the equations for t

First, solve both of the given equations for parameter t. From the first equation we find that \( t = (x - x_{1}) / (x_{2} - x_{1}) \). From the second equation we find that \( t = (y - y_{1}) / (y_{2} - y_{1}) \).

Step 2: Set the expressions equal to each other

Following the elimination of t, set the right-hand sides of both equations equal to each other, because both are equal to t. This gives us \( (x - x_{1}) / (x_{2} - x_{1}) = (y - y_{1}) / (y_{2} - y_{1}) \).

Step 3: Rearrange equation and simplify

Now, cross-multiply and afterwards simplify the equation, to find the final solution. The cross-multiply gives us \( (x - x_{1}) (y_{2} - y_{1}) = (y - y_{1}) (x_{2} - x_{1}) \). It is the rectangular form of the given line.

Key Concepts

Parametric Equations

In mathematics, a parametric equation represents a curve or a geometric object where each point on the curve is expressed in terms of one or more parameters. For a line, these equations express the x and y coordinates as functions of a parameter, usually denoted as \(t\). This approach is especially useful when studying the path or trajectory of an object or when considering multiple variables changing at once.

A parametric equation provides a clear way to describe positions along a line with respect to a parameter. For instance:

• The equation \(x = x_1 + t(x_2 - x_1)\) represents how the x-coordinate changes as \(t\) varies.
• Similarly, \(y = y_1 + t(y_2 - y_1)\) describes the change in the y-coordinate.

By adjusting \(t\), you can find different points on a line that pass through \((x_1, y_1)\) and \((x_2, y_2)\). Here, \(t = 0\) gives you \((x_1, y_1)\) and \(t = 1\) gives \((x_2, y_2)\), capturing the entire segment of the line between these two points.

Parametric equations are widely used in fields such as engineering, physics, and computer graphics. They help in handling curves and surfaces in a comprehensible and flexible manner.

Elimination of Parameter

Eliminating the parameter in parametric equations means converting these equations into a single equation in terms of only x and y, often referred to as a rectangular or Cartesian equation. The process essentially removes the parameter \(t\) to express the relationship between x and y directly.

To eliminate the parameter, follow these general steps:

• First, solve each of the parametric equations for the parameter \(t\).
• In the given problem, we find \(t\) as \((x - x_1) / (x_2 - x_1)\) and \((y - y_1) / (y_2 - y_1)\).
• Set these two expressions for \(t\) equal to each other.

This means the equations \((x - x_1) / (x_2 - x_1) = (y - y_1) / (y_2 - y_1)\) hold true, since both are equal to \(t\).

The last step is rearranging and simplifying the resultant equation, often involving cross-multiplying to clear fractions. Here, you obtain: \((x - x_1)(y_2 - y_1) = (y - y_1)(x_2 - x_1)\). This gives the line's standard form without any parameters.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves representing geometric figures and solving problems using a coordinate system. It provides a comprehensive framework to analyze geometrical shapes and understand their properties through algebraic equations.

In the context of the given problem, coordinate geometry allows us to express a line through points \((x_1, y_1)\) and \((x_2, y_2)\) based on their coordinates. Using parameters, we initially describe the movement along the line but convert it to a more general x and y relation using rectangular equations.

Some key aspects of coordinate geometry include:

• Understanding the form and equation of lines, e.g., line equations can predict intersections, slopes, and orientation.
• Utilizing coordinates to calculate distances, midpoints, and centroids of geometric figures.
• Applying algebraic methods to solve geometric problems, like finding intersection points or computing areas.

Using coordinate geometry, geometric problems become a series of algebraic tasks, which is invaluable in fields like physics and engineering. It harmonizes the study of shapes with the precision of algebra, providing versatile problem-solving skills.