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Question: How do you convert \[{{x}^{2}}-{{y}^{2}}=5\] in polar form?...

How do you convert x2−y2=5{{x}^{2}}-{{y}^{2}}=5 in polar form?

Explanation

Solution

This question is from the topic of polar system. For solving this question, we should know the relations between x and y with the term r. First, we will know the relations between them. After that we convert the equation in polar form.

Complete step by step answer:
Let us solve this question.
In this question, we have asked to find the conversion of the term x2−y2=5{{x}^{2}}-{{y}^{2}}=5 into polar form.
So, for the polar form we can see the below figure.

The relations we can see are y=rsin⁡θy=r\sin \theta and x=rcos⁡θx=r\cos \theta
So, after putting the values of x and y we can write in the equation x2−y2=5{{x}^{2}}-{{y}^{2}}=5 as
(rcos⁡θ)2−(rsin⁡θ)2=5{{\left( r\cos \theta \right)}^{2}}-{{\left( r\sin \theta \right)}^{2}}=5
The above equation can also be written as
⇒r2cos⁡2θ−r2sin⁡2θ=5\Rightarrow {{r}^{2}}{{\cos }^{2}}\theta -{{r}^{2}}{{\sin }^{2}}\theta =5
We can take r2{{r}^{2}} as common from the left side of the above equation. Hence, we will get
⇒r2(cos⁡2θ−sin⁡2θ)=5\Rightarrow {{r}^{2}}\left( {{\cos }^{2}}\theta -{{\sin }^{2}}\theta \right)=5
We are going to use a formula here in the above equation. The formula is cos⁡2θ=cos⁡2θ−sin⁡2θ\cos 2\theta ={{\cos }^{2}}\theta -{{\sin }^{2}}\theta .
Hence, we can write the above equation as
⇒r2(cos⁡2θ)=5\Rightarrow {{r}^{2}}\left( \cos 2\theta \right)=5
We can write the above equation as
⇒r2=5cos⁡2θ\Rightarrow {{r}^{2}}=\dfrac{5}{\cos 2\theta }
Taking square roots on both sides of the equation, we get
⇒r=5cos⁡2θ\Rightarrow r=\sqrt{\dfrac{5}{\cos 2\theta }}

Hence, the conversion of x2−y2=5{{x}^{2}}-{{y}^{2}}=5 in polar form will be r=5cos⁡2θr=\sqrt{\dfrac{5}{\cos 2\theta }}.

Note: For solving this type of question, we should know how to convert Cartesian form to polar form. Always remember that in polar form, we have only two variables which are r and θ\theta (theta). And in Cartesian form we have only two variables which are x and y. The relation between them is given below:
y=rsin⁡θy=r\sin \theta and x=rcos⁡θx=r\cos \theta .
We should know some formulas of trigonometry. They are very useful in various types of questions. One formula of trigonometry we have used here is cos⁡2θ=cos⁡2θ−sin⁡2θ\cos 2\theta ={{\cos }^{2}}\theta -{{\sin }^{2}}\theta . So, don’t forget the formulas of trigonometry.