Convert X2 Y24 to a polar Equation Convert r 8cos to a Carte

Convert X^2 +Y^2=4 to a polar Equation. Convert r= -8cos to a Cartesian equation. Convert r= 4/1+sin to a Cartesian equation. Can you please show the procedure, not just the answer.

Solution

We have some rules in cartesian to polar conversion:-

x= rcos -------------------------(1)

y= rsin-----------------------------(2)

x^2 +y^2 = r^2--------------------(3)

a) we have given

X^2 +Y^2=4

now from rule 3 written above we have x^2+y^2 = r^2, using this we have

r^2 = 4

=> r= 2, r= -2

b) r= -8cos

from equation 1 we have x= rcos => cos = x/r hence we have

r= -8*x/r

=> r^2 = -8x

=> x^2 +y^2 = -8x

=> x^2 +y^2 +8x = 0

c) r= 4/1+sin

cross multiply we get,

r(1+sin ) = 4

=> r+ rsin = 4

now r= sqrt(x^2+y^2 and rsin= y, hence we have

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

=> sqrt(x^2+y^2) + y- 4=0