Question

Transform the following polar equation into an equation in rectangular coordinates: r=2 cos theta A. x + y = 2 B. y = 2x C. x = 2 D. (x-1)^2+y^2=1

Answer 1

`r=2\cos\theta`

`r^2=2r\cos\theta`

`\implies x^2+y^2=2x`

`x^2-2x+y^2=0`

`x^2-2x+1+y^2=1`

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

Answer 2

The correct option is D. (x - 1)² + y² = 1

Explanation:

Given the equation in polar coordinates : r = 2 cosθ

To change the given equation in rectangular coordinates, we use the relation : x = r cosθ and y = r sinθ

⇒ x² + y² = r²

Now, r = 2 cosθ

Multiplying by r on both the sides

⇒ r² = 2r cosθ

⇒ x² + y² = 2x

⇒ x² - 2x + y² = 0

Making the variable x, the complete square by adding 1 on both the sides

⇒ x² - 2x + 1 + y² = 0 + 1

⇒ (x - 1)² + y² = 1

Hence, This is our required rectangular coordinate form of the given equation.

Therefore, The correct option is D. (x - 1)² + y² = 1