Question

How do you convert the rectangular equation to polar form $x = 4$ ?

Answer 1

In this problem, we are given the rectangular equation of some $x$ value and we are asked to convert the rectangular equation to polar form. To convert from rectangular equation to polar equation, we need to write $x,y$ as in polar coordinates $\left( {r,\theta } \right)$ and then we have to substitute the given rectangular equation.

Complete step-by-step solution:
The given rectangular equation is $x = 4$ .
Then we use $x = r\cos \theta$ and $y = r\sin \theta$ .
But we have given only the rectangular equation is $x = 4$
Substitute the given rectangular equation in $x = r\cos \theta$ , we get,
$\Rightarrow r\cos \theta = 4$ ………..…. (1)
Now let’s divide the equation (1) by $\cos \theta$ on both the sides, we get,
$\Rightarrow \dfrac{{r\cos \theta }}{{\cos \theta }} = \dfrac{4}{{\cos \theta }}$ ……….…. (2)
Then in the right-hand side of equation (2), $\cos \theta$ in both numerator and denominator get canceled by each other, we get,
$\Rightarrow r = \dfrac{4}{{\cos \theta }}$
Therefore, $r = \dfrac{4}{{\cos \theta }}$ this is the equation in polar form.

So, $r = \dfrac{4}{{\cos \theta }}$ this is our required answer.

Additional Information: A rectangular equation or an equation in rectangular form is an equation composed of variables like $x$ and $y$ which can be graphed on a regular Cartesian plane. A polar equation is an equation that describes a relation between $r$ and $\theta$ , where $r$ represents the distance from pole to a point on a curve, and $\theta$ represents the clockwise angle made by a point on a curve, the pole, and the positive $x$ - axis.

Note: Here, in this problem, we converted the rectangular equation into polar equation. And we have given a rectangular equation as a linear equation contains only one variable which is $x$ . So, we used the polar coordinate $\left( {r,\theta } \right)$ for $x$ -coordinate only and from that we converted the rectangular equation into polar equation.