Find the cartesian form of r(1 + sin θ ) =a

Question

rational · Answer

$x=r\cos\theta$
$y=r\sin\theta$

$x^2+y^2=r^2$

rational · Answer

$$r(1+\sin\theta)=a$$
$$r+r\sin\theta=a$$
$$r+y=a$$
$$r=a-y$$

square both sides and replace $r^2$ by $x^2+y^2$

roberts.spurs19 · Answer

Ah ok thank you! 
So I think I end up with $$y = \sqrt{a ^{2}-2x ^{2}}$$

rational · Answer

doesn't look correct
how did u get that ?

roberts.spurs19 · Answer

i did
$$r^2 = a^2 - x^2$$

$$x^2 +y^2 = a^2 -x^2$$

$$y^2 = a^2 -2x^2$$
which then gave me
y = $$\sqrt{a^2 - x^2}$$

rational · Answer

Lets take it step by step
you have : $$r=a-y$$

squaring both sides gives you
$$r^2=(a-y)^2$$

right ?

roberts.spurs19 · Answer

thank you! yep

rational · Answer

good, simply replace left hand side by $x^2+y^2$. you get
$$x^2+y^2=(a-y)^2$$

rational · Answer

expand right hand side and cancel y^2

$$x^2+y^2=a^2-2ay+y^2$$

$$x^2=a^2-2ay$$

rational · Answer

which is same as
$$x^2+2ay=a^2$$

roberts.spurs19 · Answer

ok then to make y the subject it would be $$y= \frac{ a^2 - x^2} { 2a }$$

rational · Answer

looks good!

roberts.spurs19 · Answer

Thank you so much for your help!!

rational · Answer

yw