Question

How do you convert $r=\dfrac{4}{1-\cos \theta }$ to rectangular form.

Answer 1

In this problem, we have to convert the given polar form into a rectangular form. We should know about the rectangular coordinates, the polar coordinates and coordinate conversion equations to convert from a polar to a rectangular form. We change the polar form to rectangular form step by step, to get a rectangular form.

Complete step by step answer:
We also know that the rectangular coordinates are $\left( x,y \right)$ and the polar coordinates are $\left( r,\theta \right)$.
We know that the coordinate conversion equations are,

& x=r\cos \theta ......(1) \\\ & y=r\sin \theta ......(2) \\\ \end{aligned}$$ Now we can apply Pythagoras’ Theorem for the above equations, we get $$\begin{aligned} & \Rightarrow {{x}^{2}}+{{y}^{2}}={{r}^{2}}{{\cos }^{2}}\theta +{{r}^{2}}{{\sin }^{2}}\theta \\\ & \Rightarrow {{x}^{2}}+{{y}^{2}}={{r}^{2}}\text{ }\because \text{si}{{\text{n}}^{2}}\theta +{{\cos }^{2}}\theta =1 \\\ \end{aligned}$$ Now we can square on both sides, we get $$r=\sqrt{{{x}^{2}}+{{y}^{2}}}$$ …… (3) We know that the given polar form is, $$r=\dfrac{4}{1-\cos \theta }$$ We can multiply $$1-\cos \theta $$ on both sides, we get $$\Rightarrow r\left( 1-\cos \theta \right)=\dfrac{4\left( 1-\cos \theta \right)}{\left( 1-\cos \theta \right)}$$ Now we can cancel the similar terms, we get $$\Rightarrow r-r\cos \theta =4$$ Now we can substitute the equation (1) and (3) in the above step, we get $$\Rightarrow \sqrt{{{x}^{2}}+{{y}^{2}}}-x=4$$ We can now add x on both sides, we get $$\Rightarrow \sqrt{{{x}^{2}}+{{y}^{2}}}=4+x$$ We can now square on both sides to get, $$\begin{aligned} & \Rightarrow {{x}^{2}}+{{y}^{2}}={{\left( x+4 \right)}^{2}} \\\ & \Rightarrow {{x}^{2}}+{{y}^{2}}={{x}^{2}}+8x+16 \\\ \end{aligned}$$ Now we can cancel the similar terms on both sides, we get $$\Rightarrow {{y}^{2}}=8x+16$$ **Therefore, the rectangular form is $${{y}^{2}}=8x+16$$.** **Note:** Students make mistakes while writing the correct equation to convert to rectangular form. We should know about the rectangular coordinates, the polar coordinates and coordinate conversion equations to convert from a polar to a rectangular form. We have used Pythagoras’ Theorem to convert the polar coordinates to the rectangular coordinates which should be concentrated.