Question

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

Answer 1

Hint: 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,
\[\begin{align}
  & x=r\cos \theta ......(1) \\
 & y=r\sin \theta ......(2) \\
\end{align}\]
 Now we can apply Pythagoras’ Theorem for the above equations, we get
\[\begin{align}
  & \Rightarrow {{x}^{2}}+{{y}^{2}}={{r}^{2}}{{\cos }^{2}}\theta +{{r}^{2}}{{\sin }^{2}}\theta \\
 & \Rightarrow {{x}^{2}}+{{y}^{2}}={{r}^{2}}\text{ }\because \text{si}{{\text{n}}^{2}}\theta +{{\cos }^{2}}\theta =1 \\
\end{align}\]
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,
\[\begin{align}
  & \Rightarrow {{x}^{2}}+{{y}^{2}}={{\left( x+4 \right)}^{2}} \\
 & \Rightarrow {{x}^{2}}+{{y}^{2}}={{x}^{2}}+8x+16 \\
\end{align}\]
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.