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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 12.1.95a
Chapter 12, Problem 12.1.95a

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.


a. x = 1 + s, y = 2s and x = 1 + 2t, y = 3t

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Identify the parametric equations of the two lines. The first line is given by \(x = 1 + s\) and \(y = 2s\), and the second line is given by \(x = 1 + 2t\) and \(y = 3t\), where \(s\) and \(t\) are parameters.
To check if the lines intersect, set the \(x\) and \(y\) coordinates equal to each other because at the point of intersection both lines share the same coordinates. So, solve the system: \(1 + s = 1 + 2t\) and \(2s = 3t\).
From the first equation, simplify to find a relationship between \(s\) and \(t\): \(s = 2t\).
Substitute \(s = 2t\) into the second equation \(2s = 3t\) to get \(2(2t) = 3t\), which simplifies to \(4t = 3t\).
Analyze the resulting equation to determine if there is a solution for \(t\). If a solution exists, use it to find \(s\) and then substitute back into the parametric equations to find the coordinates of the intersection point. If no solution exists, conclude that the lines are parallel and do not intersect.

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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 as t or s. 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 Parallelism of Lines

Two lines are parallel if their direction vectors are scalar multiples of each other. By comparing the coefficients of the parameters in the parametric equations, one can check if the lines have the same direction, indicating parallelism.
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Slopes of Tangent Lines

Finding the Point of Intersection

If lines are not parallel, their point of intersection can be found by equating their parametric expressions and solving the resulting system of equations for the parameters. Substituting these values back gives the coordinates of the intersection point.
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Related Practice
Textbook Question

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Textbook Question

Navigating A plane is 150 miles north of a radar station, and 30 minutes later it is 60 degree east of north at a distance of 100 miles from the radar station. Assume the plane flies on a straight line and maintains constant altitude during this 30-minute period.

a. Find the distance traveled during this 30-minute period.

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Textbook Question

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Textbook Question

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a. Express the region formed by the channel as a set in polar coordinates.

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Textbook Question

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a. Determine dy/dx in terms of t and evaluate it at the given value of t.


x = 2 + 4t, y = 4 − 8t; t = 2

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Textbook Question

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

b. On every ellipse, there are exactly two points at which the curve has slope s, where s is any real number.