transform the following polar equation into an equation in rectangular coordinates r=-4 sin theta.A. x=-4y B. x+y= -4 C. y= -4 D. x^2+(y+2)^2=4

Expert Answers
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You should remeber that you may convert the polar coordinates into rectangular coordinates using the following formulas such that:

`sin theta = y/r => y = r sin theta`

`cos theta = x/r => x = r cos theta`

`x^2 + y^2 = r^2`

The problem provides the information that `r = - 4 sin theta` , hence, multiplying by r both sides yields:

`r^2 = -4r sin theta => r^2 = -4y`

Substituting `x^2 + y^2`  for `r^2`  yields:

`x^2 + y^2 = -4y`

You should move `-4y`  to the left side such that:

`x^2 + y^2 + 4y = 0`

You need to complete the square `y^2 + 4y`  adding 4 both sides such that:

`x^2 + y^2 + 4y + 4 = 4`

You need to convert the expansion `y^2 + 4y + 4`  into t he square of binomial such that:

`x^2 + (y+2)^2 = 4`

Hence, converting the polar form of equation into the rectangular form yields `x^2 + (y+2)^2 = 4,`  thus you need to select the answer D.