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

Verified step by step guidance

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: \(\boxed{r = 2 (a \cos \theta + b \sin \theta)}\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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.

Recommended video:

05:32

Intro to Polar Coordinates

Related Practice

Textbook Question

13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.

x²/9 + y²/4 = 1

views

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 upper half of the parabola x=y ², originating at (0, 0)

views

Textbook Question

25–30. Converting coordinates Express the following polar coordinates in Cartesian coordinates.

(2, 7π/4)

views

Textbook Question

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

The region inside the lemniscate r² = 6 sin 2θ

views

Textbook Question

15–30. Working with parametric equations Consider the following parametric equations.