Textbook Question
Theory and Examples
Tangents Find equations for the tangents to the circle (x − 2)² + (y − 1)² = 5 at the points where the circle crosses the coordinate axes.
Ch. 11 - Parametric Equations and Polar Coordinates
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 11, Problem 11.7.55
Circles
Sketch the circles in Exercises 53–56. Give polar coordinates for their centers and identify their radii.
r = −2 cos θ
Verified step by step guidance1
Recall that the general form of a circle in polar coordinates can often be written as \(r = a \cos \theta + b \sin \theta + c\). In this problem, the equation is \(r = -2 \cos \theta\), which can be analyzed to find the center and radius of the circle.
Rewrite the equation \(r = -2 \cos \theta\) by factoring out the negative sign: \(r = - (2 \cos \theta)\). This suggests the circle is related to the cosine function with a negative coefficient.
Convert the polar equation to Cartesian coordinates using the relationships \(x = r \cos \theta\) and \(y = r \sin \theta\). Multiply both sides of the equation by \(r\) to get \(r^2 = -2 r \cos \theta\).
Substitute \(r^2 = x^2 + y^2\) and \(r \cos \theta = x\) into the equation to obtain \(x^2 + y^2 = -2x\). Rearrange this to \(x^2 + 2x + y^2 = 0\).
Complete the square for the \(x\)-terms: \(x^2 + 2x + 1 + y^2 = 1\). This can be written as \((x + 1)^2 + y^2 = 1\), which represents a circle with center at \((-1, 0)\) in Cartesian coordinates and radius \(1\). Convert the center to polar coordinates by calculating \(r = \sqrt{(-1)^2 + 0^2}\) and \(\theta = \arctan(0 / -1)\).
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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 radius and an angle (r, θ) from the origin. Unlike Cartesian coordinates, the position depends on the distance from the origin and the direction, making it useful for curves like circles defined by r as a function of θ.
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Intro to Polar Coordinates
Equation of a Circle in Polar Form
A circle in polar coordinates can often be expressed as r = a ± b cos θ or r = a ± b sin θ. These forms correspond to circles with centers offset from the origin, and understanding how to convert or interpret these equations helps identify the circle's center and radius.
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6:50Convert Equations from Polar to Rectangular
Conversion Between Polar and Cartesian Coordinates
To analyze or sketch polar curves, converting between polar (r, θ) and Cartesian (x, y) coordinates is essential. Using x = r cos θ and y = r sin θ allows one to rewrite the polar equation in Cartesian form, facilitating identification of geometric properties like the center and radius of a circle.
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Related Practice
Textbook Question
Shifting Conic Sections
Find the center, foci, vertices, asymptotes, and radius, as appropriate, of the conic sections in Exercises 57-68.
9x² + 6y² + 36y = 0
Textbook Question
Finding Cartesian from Parametric Equations
Exercises 1–18 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.
x = 2 sinh t, y = 2 cosh t, −∞<t<∞
Textbook Question
Ellipses
Exercises 25 and 26 give information about the foci and vertices of ellipses centered at the origin of the xy−plane. In each case, find the ellipse's standard−form equation from the given information.
Foci: ( ±√2, 0) Vertices: (±2,0)
Textbook Question
Parabolas
Exercises 9-16 give equations of parabolas. Find each parabola's focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch.
x = −3y²
Textbook Question
Parabolas
Exercises 9-16 give equations of parabolas. Find each parabola's focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch.
x² = 6y