Ch.12 - Parametric and Polar Curves

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problem 12.2.4

Chapter 12, Problem 12.2.4

What is the polar equation of a circle of radius √(a²+b²) centered at (a, b)?

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Recall that the Cartesian equation of a circle centered at \((a, b)\) with radius \(r\) is given by \((x - a)^2 + (y - b)^2 = r^2\). In this problem, the radius is \(\sqrt{a^2 + b^2}\), so the equation becomes \((x - a)^2 + (y - b)^2 = a^2 + b^2\).

Express the Cartesian coordinates \(x\) and \(y\) in terms of polar coordinates \(r\) and \(\theta\) using the relations \(x = r \cos \theta\) and \(y = r \sin \theta\). Substitute these into the circle equation to get \((r \cos \theta - a)^2 + (r \sin \theta - b)^2 = a^2 + b^2\).

Expand the squares on the left-hand side: \((r \cos \theta)^2 - 2 a r \cos \theta + a^2 + (r \sin \theta)^2 - 2 b r \sin \theta + b^2 = a^2 + b^2\).

Combine like terms and simplify. Notice that \((r \cos \theta)^2 + (r \sin \theta)^2 = r^2\), and \(a^2 + b^2\) appears on both sides, so they cancel out. This leaves the equation \(r^2 - 2 a r \cos \theta - 2 b r \sin \theta = 0\).

Factor out \(r\) from the terms involving it: \(r^2 = 2 r (a \cos \theta + b \sin \theta)\). Since \(r \neq 0\), divide both sides by \(r\) to get the polar equation of the circle: \(\boxed{r = 2 (a \cos \theta + b \sin \theta)}\).

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Key Concepts

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

Polar Coordinates

Polar coordinates represent points in the plane using a distance from the origin (r) and an angle (θ) from the positive x-axis. Unlike Cartesian coordinates (x, y), polar coordinates express location based on radius and direction, which is essential for converting equations of curves into polar form.

Equation of a Circle in Cartesian Coordinates

A circle centered at (a, b) with radius R has the Cartesian equation (x - a)² + (y - b)² = R². Understanding this standard form is crucial before converting it into polar coordinates, as it provides the geometric definition and parameters of the circle.

Conversion Between Cartesian and Polar Coordinates

To convert Cartesian equations to polar form, use x = r cos θ and y = r sin θ. Substituting these into the Cartesian equation allows rewriting the curve in terms of r and θ, enabling the derivation of the polar equation of the circle.

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Intro to Polar Coordinates

Related Practice

Textbook Question

Air drop—inverse problem A plane traveling horizontally at 100 m/s over flat ground at an elevation of 4000 m must drop an emergency packet on a target on the ground. The trajectory of the packet is given by

x = 100t, y = −4.9t² + 4000, t ≥ 0

where the origin is the point on the ground directly beneath the plane at the moment of the release. How many horizontal meters before the target should the packet be released in order to hit the target?

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Textbook Question

63–74. Arc length of polar curves Find the length of the following polar curves.

{Use of Tech} The complete limaçon r=4−2cosθ

Textbook Question

45–60. Areas of regions Find the area of the following regions.

The region common to the circles r = 2 sin θ and r = 1

Textbook Question

31–36. Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways.

(1, √3)

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Textbook Question

23–24. Radar Airplanes are equipped with transponders that allow air traffic controllers to see their locations on radar screens. Radar gives the distance of the plane from the radar station (located at the origin) and the angular position of the plane, typically measured in degrees clockwise from north.

A plane is 50 miles from a radar station at an angle of 10 dgeree clockwise from north. Find polar coordinates for the location of the plane.

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Textbook Question

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.

The left half of the parabola y=x ² +1, originating at (0, 1)

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