Skip to main content
Ch. 11 - Parametric Equations and Polar Coordinates
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblem 11.PE.33
Chapter 11, Problem 11.PE.33

Cartesian to Polar Equations


Find polar equations for the circles in Exercises 33–36. Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations.


x² + y² + 5y = 0

Verified step by step guidance
1
Start with the given Cartesian equation of the circle: \(x^{2} + y^{2} + 5y = 0\).
Rewrite the equation to complete the square for the \(y\) terms. Group \(y^{2} + 5y\) together: \(x^{2} + (y^{2} + 5y) = 0\).
Complete the square for \(y^{2} + 5y\) by adding and subtracting \(\left(\frac{5}{2}\right)^{2} = \frac{25}{4}\) inside the equation: \(x^{2} + \left(y^{2} + 5y + \frac{25}{4}\right) - \frac{25}{4} = 0\).
Rewrite the equation as \(x^{2} + \left(y + \frac{5}{2}\right)^{2} = \frac{25}{4}\), which represents a circle with center at \((0, -\frac{5}{2})\) and radius \(\frac{5}{2}\).
Convert to polar coordinates using \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\). Substitute these into the equation: \(\left(r \cos{\theta}\right)^{2} + \left(r \sin{\theta} + \frac{5}{2}\right)^{2} = \frac{25}{4}\). This is the polar form of the circle's equation.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
Was this helpful?

Key Concepts

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

Conversion between Cartesian and Polar Coordinates

This concept involves expressing points from the Cartesian coordinate system (x, y) in terms of polar coordinates (r, θ), where x = r cos θ and y = r sin θ. Understanding this conversion is essential to rewrite Cartesian equations in polar form.
Recommended video:
05:32
Intro to Polar Coordinates

Equation of a Circle in Cartesian Coordinates

A circle in Cartesian form is typically expressed as (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Recognizing and rewriting the given equation into this standard form helps identify the circle’s center and radius.
Recommended video:
05:32
Intro to Polar Coordinates

Polar Form of a Circle Equation

After converting Cartesian coordinates to polar, the circle’s equation can be expressed in terms of r and θ. This often involves substituting x = r cos θ and y = r sin θ and simplifying to isolate r, which helps in sketching and understanding the circle in polar coordinates.
Recommended video:
6:50
Convert Equations from Polar to Rectangular
Related Practice
Textbook Question

Length in Polar Coordinates


Find the lengths of the curves given by the polar coordinate equations in Exercises 51–54.


r = √(1 + cos 2θ), −π/2 ≤ θ ≤ π/2

1
views
Textbook Question

Graphing Conic Sections


Sketch the parabolas in Exercises 55–58. Include the focus and directrix in each sketch.


y² = −(8/3)x

Textbook Question

Lengths of Curves


Find the lengths of the curves in Exercises 13–19.


x = 5 cos t − cos 5t, y = 5 sin t − sin 5t, 0 ≤ t ≤ π/2

1
views
Textbook Question

Finding Parametric Equations


Find parametric equations and a parameter interval for the motion of a particle that starts at (a, 0) and traces the circle x²+y²=a².


a. once clockwise.


(There are many ways to do these, so your answers may not be the same as the ones at the back of the text.)

Textbook Question

Area in Polar Coordinates


Find the areas of the regions in the polar coordinate plane described in Exercises 47–50.


Inside the cardioid r = 2(1 + sin θ) and outside the circle r = 2 sin θ

1
views
Textbook Question

Cycloid


a. Find the length of one arch of the cycloid x = a(t − sin t), y = a(1 − cos t).

1
views