Textbook Question
Area of a sector of a hyperbola: Consider the region R bounded by the right branch of the hyperbola x²/a² - y²/b² = 1 and the vertical line through the right focus
a. What is the area of R?
89
views
Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
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
Verified step by step guidance1
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.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4mWas this helpful?
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.
Recommended video:
Guided course
08:02
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.
Recommended video:
05:13Slopes 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.
Recommended video:
04:50Critical Points
Related Practice
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.
22
views
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).
a. Find the area Aₙ of Rₙ.
76
views
Textbook Question
Channel flow Water flows in a shallow semicircular channel with inner and outer radii of 1 m and 2 m (see figure). At a point P(r,θ) in the channel, the flow is in the tangential direction (counterclock wise along circles), and it depends only on r, the distance from the center of the semicircles.
a. Express the region formed by the channel as a set in polar coordinates.
52
views
Textbook Question
67–72. Derivatives Consider the following parametric curves.
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
50
views
Textbook Question
Volume of a hyperbolic cap Consider the region R bounded by the right branch of the hyperbola x²/a² - y²/b² = 1 and the vertical line through the right focus.
a. What is the volume of the solid that is generated when R is revolved about the x-axis?
40
views