Question

Convert the polar equation to rectangular form
`r = 2 cos theta + 2 sin theta`

A.
`(x-1)^2-(y-1)^2=2`

B.
`(x+1)^2-(y+1)^2=2`

C.
`(x-2)^2-(y-2)^2=1`

D.
`(x+2)^2-(y+2)^2=1`

Answer 1

A. (x - 1)² + (y - 1)² = 2

This is the equation of circle with center at (1, 1) and radius √2

Explanation:

We can use the following trigonometric identities to convert the given polar equation to rectangular form:

cosθ = x / r

sinθ = y / r

Substituting these identities into the given polar equation, we get:

r = 2cosθ + 2sinθ = 2(x / r) + 2(y / r)

Multiplying both sides by r, we get:

r² = 2x + 2y

We can also use the Pythagorean identity to express r² in terms of x and y:

r² = x² + y²

Substituting this expression into the previous equation, we get:

x² + y² = 2x + 2y

x²+y² = 2y+2x

x²-2x+y2-2y=0

(x²-2x+1)+(y²-2y+1)=2

(x-1)²+(y-1)² = (√2)²

This is the equation of circle with center at (1, 1) and radius √2

OR

Completing the square for both x and y terms, we get:

(x - 1)² - 1 + (y - 1)² - 1 = 0

Simplifying, we get:

(x - 1)² + (y - 1)² = 2

Therefore, the rectangular form of the polar equation r = 2cosθ + 2sinθ is:

A. (x - 1)² + (y - 1)² = 2

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