Question

Convert r =-72/12+6 sin theta to Cartesian form.

Answer 1

For converting polar to cartesian form we know
$r = \sqrt{x^2 + y^2}$
$x = rcos \theta , y = rsin \theta$

Given equation is

$r = \frac{-72}{12} + 6sin \theta$

We can simplify it as
$r = -6 + 6sin \theta$

Now we can write it as
$r^2 = -6r + 6rsin \theta$

Now we can use
$r^2 = x^ 2+ y^2 , rsin \theta = y , r = \sqrt{x^2 + y^2}$
So equation we can write it as
$x^2 + y^2 = -6 \sqrt{x^2 + y^2} + 6y$
$x^2 + y^2 - 6y = -6 \sqrt{x^2 + y^2}$
On squaring both sides
$(x^2 + y^2 - 6y)^2 = (-6 \sqrt{x^2 + y^2})^2$
$x^4 + y^4 + 36y^2 +2x^2y^2 -12y^3 -12x^2y = 36(x^2 + y^2)$
$x^4 + y^4 +36y^2 + 2x^2y^2 -12y^3 -12x^2y = 36x^2 + 36y^2$
$x^4 + y^4 + 36y^2 +2x^2y^2 -12y^3-12x^2y - 36x^2 - 36y^2 = 0$
$x^4 + y^4 -12y^3 + 2x^2y^2 - 12x^2y - 36x^2 = 0$

Answer 2

Answer:

The answer is b. $4x^2+3y^2-24y-144=0$

Step-by-step explanation:

The equation is:

$r=\frac{-72}{12+6\sin(\theta)}$

Cross multiply;:

$r(12+6\sin(\theta)=-72$

Expand;:

$12r+6r\sin(\theta)=-72$

Divide through by 6;:

$2r+r\sin(\theta)=-12$

$2r=-12-r\sin(\theta)$

Substitute;:

$y=r\sin(\theta), r=\sqrt{x^2+y^2}$

$2\sqrt{x^2+y^2}=-12-y$

Square both sides;:

$(2\sqrt{x^2+y^2})^2=(-12-y)^2$

$4(x^2+y^2)=144+24y+y^2$

$4x^2+4y^2=144+24y+y^2$

Simplify;:

$4x^2+3y^2-24y-144=0$