Parametrically Defined Circles and Lines - AP PreCalculus Study Guide

The Core Idea: Parametrically Defined Circles and Lines

In mathematics, we often describe curves using a single equation relating $x$ and $y$, such as $y=mx+b$ or $(x-h{)}^2+(y-k{)}^2=r^2$. Parametric equations offer a different and powerful perspective. Instead of defining a curve based on the relationship between its horizontal and vertical coordinates, we introduce a third variable, called a parameter, often denoted by $t$. Both the $x$ and $y$ coordinates are defined independently as functions of this parameter $t$. This approach, $x=x(t)$ and $y=y(t)$, is particularly useful for modeling motion, where $t$ can represent time.

This topic focuses on how to use this parametric framework to describe two fundamental geometric shapes: lines and circles. For a line, the parametric equations will show a starting point and a constant rate of change in the $x$ and $y$ directions. For a circle, the equations will use trigonometric functions, sine and cosine, to trace a path of constant distance from a central point. By defining a specific interval for the parameter $t$, we can describe not just a full line or circle, but also specific line segments or circular arcs, allowing us to model paths with clear starting and ending points.

Key Formulas

Parametric Form of a Line

The path of an object moving along a straight line can be described by the following pair of parametric equations:

$$x(t)=x_0+at$$

$$y(t)=y_0+bt$$

• $(x_0,y_0)$: This represents the initial position of the object at time $t=0$.

• $a$: This is the constant rate of change of the horizontal position ($x$) with respect to the parameter $t$.

• $b$: This is the constant rate of change of the vertical position ($y$) with respect to the parameter $t$.

• The pair of values $⟨a,b⟩$ can be thought of as a direction vector that determines the slope and direction of the line.

Parametric Form of a Circle

A circle with center `(h, k)and radius $r can be described by the following pair of parametric equations:

$$x(t)=h+r\cos (t)$$

$$y(t)=k+r\sin (t)$$

• $(h,k)$: This is the center of the circle.

• $r$: This is the radius of the circle ($r>0$).

• $t$: The parameter $t$ typically represents the angle in radians. As $t$ increases from $0$ to $2\pi$, the point $(x(t),y(t))$ traces the circle one full time, usually in a counter-clockwise direction.

Understanding The Parameter Interval

A critical feature of parametric equations is the ability to define the domain of the parameter $t$. This allows us to trace specific portions of a curve rather than the entire infinite line or complete circle. This is known as restricting the parameter.

For example, consider a line defined by $x(t)=1+2t$ and $y(t)=3+4t$.

• If there is no restriction on $t$, the equations describe the entire line.

• If the parameter is restricted to the interval $0\le t\le 1$, the equations describe a line segment.

  • The starting point (at $t=0$) is $(x(0),y(0))=(1,3)$.

  • The ending point (at $t=1$) is $(x(1),y(1))=(3,7)$.

Similarly, for a circle defined by $x(t)=5\cos (t)$ and $y(t)=5\sin (t)$:

• If the parameter is defined on the interval $0\le t\le 2\pi$, the equations describe the full circle.

• If the parameter is restricted to $0\le t\le \pi$, the equations describe a semicircle.

  • The starting point (at $t=0$) is $(5,0)$.

  • The ending point (at $t=\pi$) is $(-5,0)$.

This ability to define segments and arcs makes parametric equations essential for modeling real-world paths that have a specific start and end.

Core Concepts & Rules

• Parametric equations for a line define $x(t)$ and $y(t)$ as linear functions of the parameter $t$.

• In the linear form $x(t)=x_0+at$ and $y(t)=y_0+bt$, the point $(x_0,y_0)$ is the position of the object when the parameter $t=0$.

• The coefficients $a$ and $b$ in the linear form determine the direction of the line's path.

• Parametric equations for a circle involve the trigonometric functions $\cos (t)$ and $\sin (t)$.

• In the circular form $x(t)=h+r\cos (t)$ and $y(t)=k+r\sin (t)$, the point $(h,k)$ is the center of the circle and $r$ is its radius.

• Restricting the domain of the parameter $t$ to a closed interval, such as $[t_1,t_2]$, results in tracing only a segment of the curve, such as a line segment or a circular arc.

Step-by-Step Example 1: Writing Parametric Equations for a Line Segment

Problem: A particle starts at the point $P=(-2,3)$ when $t=0$ and moves along a straight line to the point $Q=(4,7)$ at $t=2$. Write the parametric equations that describe this motion.

Step 1: Identify the initial position $(x_0,y_0)$.

The problem states that the particle is at $(-2,3)$ when $t=0$. Therefore, we have:

$$x_0=-2$$

$$y_0=3$$

Step 2: Determine the direction components $a$ and $b$.

The general form is $x(t)=x_0+at$ and $y(t)=y_0+bt$. We know the position at $t=2$ is $(4,7)$. We can plug these values in to solve for $a$ and $b$.

For the x-coordinate:

$$x(2)=x_0+a(2)$$

$$4=-2+2a$$

$$6=2a$$

$$a=3$$

For the y-coordinate:

$$y(2)=y_0+b(2)$$

$$7=3+2b$$

$$4=2b$$

$$b=2$$

Step 3: Write the final parametric equations.

Now that we have $x_0,y_0,a,$ and $b$, we can write the equations:

$$x(t)=-2+3t$$

$$y(t)=3+2t$$

Step 4: State the parameter interval.

The motion occurs from $t=0$ to $t=2$. Therefore, the full description includes the interval:

$$x(t)=-2+3t,y(t)=3+2t,\text{for }0\le t\le 2$$

Step-by-Step Example 2: Analyzing a Parametrically Defined Circular Arc

Problem: The path of a particle is described by the parametric equations $x(t)=4+3\cos (t)$ and $y(t)=-1+3\sin (t)$, for the interval $\frac{\pi }{2}\le t\le \frac{3\pi }{2}$. Identify the shape of the path, its key features, and its starting and ending points.

Step 1: Identify the shape and its key features.

The equations are in the form $x(t)=h+r\cos (t)$ and $y(t)=k+r\sin (t)$. This indicates the path is a portion of a circle.

• By comparing terms, we can identify the center (h, k).

  • $h=4$

  • $k=-1$

  • The center is $(4,-1)$.

• By comparing terms, we can identify the radius $r$.

  • $r=3$

  • The radius is 3.

The path lies on a circle centered at $(4,-1)$ with a radius of 3.

Step 2: Determine the starting point of the path.

The path starts at the minimum value of the parameter interval, $t=\frac{\pi }{2}$. We evaluate $x(t)$ and $y(t)$ at this value.

$$x\left(\frac{\pi }{2}\right)=4+3\cos \left(\frac{\pi }{2}\right)=4+3(0)=4$$

$$y\left(\frac{\pi }{2}\right)=-1+3\sin \left(\frac{\pi }{2}\right)=-1+3(1)=2$$

The starting point is $(4,2)$.

Step 3: Determine the ending point of the path.

The path ends at the maximum value of the parameter interval, $t=\frac{3\pi }{2}$. We evaluate $x(t)$ and $y(t)$ at this value.

$$x\left(\frac{3\pi }{2}\right)=4+3\cos \left(\frac{3\pi }{2}\right)=4+3(0)=4$$

$$y\left(\frac{3\pi }{2}\right)=-1+3\sin \left(\frac{3\pi }{2}\right)=-1+3(-1)=-4$$

The ending point is $(4,-4)$.

Step 4: Synthesize the description.

The particle travels along a semicircular arc of a circle centered at $(4,-1)$ with a radius of 3. The path begins at the point $(4,2)$ (the top of the circle) and moves counter-clockwise to the point $(4,-4)$ (the bottom of the circle).

Using Your Calculator

A graphing calculator is an essential tool for visualizing parametrically defined curves and verifying your analytical work.

To graph a parametric curve (e.g., the circle from Example 2):

1. Set the Mode:

   • Press the [MODE] button.

   • Navigate down to the line that reads FUNCTION (or FUNC).

   • Use the arrow keys to highlight PARAMETRIC (or PAR) and press [ENTER].

2. Enter the Equations:

   • Press the [Y=] button. You will now see input fields for pairs of equations: $X_{1}T$, $Y_{1}T$, etc.

   • In $X_{1}T$, enter the equation for $x(t)$: $4+3cos(T)$

   • In $Y_{1}T$, enter the equation for $y(t)$: $-1+3sin(T)$

   • The $[X,T,\theta ,n]$ button will produce a $T$ when in parametric mode.

3. Set the Window:

   • Press the [WINDOW] button. This is the most critical step for parametric graphing.

   • $Tmin$ and $Tmax$: Set these to the start and end of your parameter interval. For Example 2, set $Tmin=\pi /2$ and $Tmax=3\pi /2$.

   • $Tstep$: This controls the resolution of the graph. A smaller value gives a smoother curve but takes longer to draw. A good starting point is $\pi /24$.

   • $Xmin$, $Xmax$, $Ymin$, $Ymax$: Set these to define the viewing window. For a circle with center `(4, -1)and radius 3, the x-values will range from $4-3=1 to $4+3=7$, and y-values from $-1-3=-4$ to $-1+3=2$. A good window would be $Xmin=0$, $Xmax=8$, $Ymin=-5$, $Ymax=3‘.4.\ast \ast GraphandTrace:\ast \ast -Press‘[GRAPH]‘.Youshouldseethesemicirculararcdrawnonthescreen.-Press‘[TRACE]$. You can now use the left and right arrow keys to move a cursor along the curve. Notice how the values of $T$, $X$, and $Y$ are displayed. As you press the right arrow, T` increases, and you can observe the direction of motion from the starting point to the ending point. ## AP Exam Quick Hit ### Common Question Types - **Identifying Features from Equations:** You will be given a set of parametric equations, like $x(t) = -5 + 2t and $y(t)=1-4t$, and asked to identify the initial position at $t=0$ or describe the object's path.

   • Example: "For the path defined by $x(t)=7+10\cos (t)$ and $y(t)=-2+10\sin (t)$, what are the center and radius of the circular path?"

• Writing Equations from a Description: You will be given a geometric description of a path and asked to write the corresponding parametric equations.

  • Example: "Write a set of parametric equations for a line that passes through the point $(1,6)$ at $t=0$ and the point $(3,10)$ at $t=1$."

• Analyzing Restricted Intervals: You will be given parametric equations with a specific interval for $t$ and asked to find the coordinates of the start and end points of the path.

  • Example: "A particle's motion is given by $x(t)=2t$ and $y(t)=t^2-1$ for $0\le t\le 3$. What are the initial and terminal coordinates of the particle?" (Note: While this example uses a parabola, the principle applies directly to lines and circles).

Common Mistakes

• Confusing Center and Radius: In the circle equation $x(t)=h+r\cos (t)$, students might mistakenly identify $h$ as the radius or $r$ as part of the center's coordinates. Always match the formula structure carefully.

• Sign Errors with the Center: For an equation like $x(t)=5-2\cos (t)$, the center's x-coordinate is $h=5$. A common mistake is to associate the minus sign with the center and incorrectly state $h=-5$.

• Incorrectly Calculating Direction Components: When finding the equations for a line segment from $(x_0,y_0)$ to $(x_1,y_1)$ over a time interval $\Delta t$, the direction components are $a=\frac{x_1-x_0}{\Delta t}$ and $b=\frac{y_1-y_0}{\Delta t}$. A frequent error is to forget to divide by the change in time, $\Delta t$.

• Ignoring the Parameter Interval: When asked to describe a path, students often describe the full line or circle, forgetting that a restricted interval like $0\le t\le \pi /2$ means the path is only a segment or an arc. Always check for an interval.

• Mixing Up Sine and Cosine: Swapping the sine and cosine functions in the circle equations (e.g., $x(t)=h+r\sin (t)$, $y(t)=k+r\cos (t)$) is a valid parameterization, but it changes the starting point (at $t=0$) and the direction of motion. Be sure to use the standard form unless specified otherwise.