Express the given polar equation in simplest rectangular form: $r = 4 + 3cosθ$

Question

Express the given polar equation in simplest rectangular form: $r = 4 + 3cosθ$

My attempt

Multiplying r on both sides, we get

$r^2=4r+3 rcos\theta$

since, $x=rcos\theta$ and $y=rsin\theta$ and $r^2=x^2+y^2$

we get $x^2+y^2=4r+3x$

can anyone please explain after this step..

Answer 1

$$r^2 - 4 r = 3 x$$

$$x^2 +y^2 - 4\sqrt{x^2 +y^2} = 3 x$$

which represents a cardoid.. it is better to leave it that way.

if you try to get rid of radical sign you end up with another equation representing another symmetrical cardoid, not really what you wanted to represent at the outset:

$$x^2 +y^2 + 4\sqrt{x^2 +y^2} = 3 x$$

Answer 2

Hint: for $r\neq 0$ we get $$r=4+3\cos(\theta)$$ so you will get $$\sqrt{x^2+y^2}=4+\frac{3x}{\sqrt{x^2+y^2}}$$

Answer 3

You have $r= 4+3\cos(\theta)$

$r^2 = 4r+3r\cos(\theta)$

we know that ;$x= r\cos(\theta)\qquad y = r\sin(\theta)$

$r^2 = x^2+y^2$

$\implies x^2+y^2 =\sqrt{x^2+y^2}+3x$

$\implies (x^2+y^2-3x)^2 = x^2+y^2$