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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.1.95c
Chapter 12, Problem 12.1.95c

Intersecting lines Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection.


c. x = 1 + 3s, y = 4 + 2s and x = 4 - 3t, y = 6 + 4t

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Identify the parametric equations of the two lines: Line 1 is given by \(x = 1 + 3s\), \(y = 4 + 2s\) and Line 2 is given by \(x = 4 - 3t\), \(y = 6 + 4t\), where \(s\) and \(t\) are parameters.
To check if the lines are parallel, compare their direction vectors. The direction vector of Line 1 is \(\langle 3, 2 \rangle\) and for Line 2 it is \(\langle -3, 4 \rangle\). Determine if one vector is a scalar multiple of the other.
If the direction vectors are not scalar multiples, the lines are not parallel and may intersect. To find the intersection point, set the \(x\) and \(y\) coordinates equal to each other: \(1 + 3s = 4 - 3t\) and \(4 + 2s = 6 + 4t\).
Solve the system of equations for \(s\) and \(t\). This involves rearranging the equations and using substitution or elimination methods to find values of \(s\) and \(t\) that satisfy both equations simultaneously.
Once you find \(s\) and \(t\), substitute either value back into the parametric equations of one of the lines to find the coordinates of the intersection point \(\left(x, y\right)\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Parametric Equations of Lines

Parametric equations express the coordinates of points on a line as functions of a parameter, typically denoted by s or t. Each parameter value corresponds to a unique point on the line, allowing a clear representation of lines in the plane or space.
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Parameterizing Equations

Determining Intersection of Lines

To find if two lines intersect, set their parametric equations equal and solve for the parameters. If a consistent solution exists, the lines intersect at the corresponding point; otherwise, they are parallel or skew.
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Parallel Lines and Direction Vectors

Two lines are parallel if their direction vectors are scalar multiples of each other. Comparing the coefficients of the parameters in the parametric equations helps identify parallelism without solving for intersection.
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Related Practice
Textbook Question

11–14. Working with parametric equations Consider the following parametric equations.

c. Eliminate the parameter to obtain an equation in x and y.

d. Describe the curve.


x=−t+6, y=3t−3; −5≤t≤5 

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.  


d. The point (3,π/2) lies on the graph of r=3 cos 2θ.  

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Textbook Question

Spiral arc length Consider the spiral r=4θ, for θ≥0.


c. Show that L′(θ)>0. Is L″(θ) positive or negative? Interpret your answer.

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Textbook Question

11–14. Working with parametric equations Consider the following parametric equations.

c. Eliminate the parameter to obtain an equation in x and y.

d. Describe the curve.


x=2 t,y=3t−4;−10≤t≤10 

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


d. The parametric equations x=cos t, y=sin t, for −π/2≤t≤π/2, describe a semicircle.

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Textbook Question

Regions bounded by a spiral: Let Rₙ be the region bounded by the nth turn and the (n+1)st turn of the spiral r = e⁻ᶿ in the first and second quadrants, for θ ≥ 0 (see figure).

c. Evaluate lim(n→∞) Aₙ₊₁/Aₙ.


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