Problem 40 - Calculus Volume 3

Question

The pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. $$ \begin{array}{l} x-4=5 t \\ y+2=t \end{array} $$

Step-by-Step Solution

Answer

The parametric equations represent a line.

Step 1: Identify the Form of Given Equations

We have the parametric equations:\( x = 4 + 5t \) and \( y = -2 + t \). These equations express \( x \) and \( y \) in terms of a parameter \( t \).

Step 2: Recognize the Standard Equation of a Line

Parametric equations of a line typically have the form \( x = x_0 + at \) and \( y = y_0 + bt \), where \( x_0 \) and \( y_0 \) are points on the line, and \( a \) and \( b \) are direction components of the line.

Step 3: Compare the Given Equations to Linear Form

The given equations match the standard form for a line: \( x = 4 + 5t \) (with \( a = 5 \)) and \( y = -2 + t \) (with \( b = 1 \)). Hence, these equations define a line.

Key Concepts

Types of Curves

In mathematics, curves come in various types, each with unique properties. Understanding these curves is essential, especially when working with parametric equations. Here are some common types of curves you might encounter:

• Lines: The simplest type of curve. A line is straight with constant direction and no curvature.
• Parabolas: These are curves where each point is equidistant from a fixed point (focus) and a line (directrix). They have a "U" shape.
• Circles: All points are equidistant from a central point. Circles are examples of closed curves.
• Ellipses: Similar to circles but stretched along one axis. The sum of the distances from any point on the ellipse to two foci is constant.
• Hyperbolas: Two opposing curves that resemble two mirrored parabolas. Each point's distance difference from two foci is constant.

Depending on the form of parametric equations, they can represent any of these curves. Recognizing the type helps in understanding the relationship between variables.

Lines in Geometry

Lines are fundamental objects in geometry. They extend infinitely in both directions and are characterized by having no thickness or curvature.
Lines can be described using different methods like:

• Standard Form: Defined by an equation like \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants.
• Point-Slope Form: Uses a specific point’s coordinates \((x_1, y_1)\) and the slope \(m\): \(y - y_1 = m(x - x_1)\).
• Slope-Intercept Form: Easily recognizable as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
• Vector/Parametric Form: Useful for describing lines in space using vectors and parameters, such as \(x = x_0 + at\) and \(y = y_0 + bt\).

These varying forms help us interact with lines mathematically, making it easier to analyze relationships and intersections with other geometric entities.

Equations of Lines

Equations of lines allow us to represent linear paths in algebraic form. For any line, we need to know at least one point on it and its direction. This gives us the foundation to construct its equation.
A line's equation can reveal:

• Direction: The slope indicates how steep a line is. For instance, in \(y = mx + b\), \(m\) represents the slope.
• Points on Line: Any specific point on the line can help determine its equation, such as using \((x_1, y_1)\) in a point-slope form.
• Parametric Form: Utilizes parameters to express equations, helpful for lines in three-dimensional space. Example: \(x = x_0 + at\), and \(y = y_0 + bt\).

Understanding these equations enables solving geometric problems, determining intersections, and describing paths of motion in space.