Question

Convert the given cartesian equation to a polar equation
A) X^2+y^2=-2y
B)x^2+y^2=64

Answer 1

A) r = -2sin(θ)

B) r = 8

Explanation:

You want the polar coordinate versions of these Cartesian equations:

• x² +y² = -2y
• x² +y² = 64

Conversion

The conversion between Cartesian coordinates and polar coordinates can be accomplished using the relations ...

• x = r·cos(θ)
• y = r·sin(θ)

A)

Substituting for x and y, we have ...

(r·cos(θ))² +(r·sin(θ))² = -2r·sin(θ)

Dividing by r and using the Pythagorean identity, we have ...

r = -2·sin(θ)

B)

Substituting for x and y, we have ...

(r·cos(θ))² +(r·sin(θ))² = 64

Using the Pythagorean identity sin(θ)² +cos(θ)² = 1, this becomes ...

r² = 64

Taking the square root, we have ...

r = 8 . . . . . . the equation of a circle of radius 8.

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Additional comment

You know the Cartesian equation for a circle is ...

(x -h)² +(y -k)² = r² . . . . . . . circle of radius r centered at (h, k)

The equation of (A) can be rearranged to x² +(y +1)² = 1, by adding 2y and completing the square. This is a circle of radius 1 centered at (0, -1), as shown in the graph.

The Cartesian equation of (B) is clearly a circle centered at the origin with radius 8, as shown in the graph.

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