Question

How do you convert the polar equation $r = 7sec\theta \;$ into rectangular form?

Answer 1

Hint : In this question we have to convert the polar equation into rectangular form. We know that the x coordinate in polar form is represented as $x\, = \,r\cos \theta$ and the y coordinate in the polar form is represented as $y = r\sin \theta$. So, we have to find the value of x and y using the above equation.

Complete step-by-step answer :
We know that we can write the x and y coordinate of any locus in the form of theta or we can say polar form.
Therefore,
$\Rightarrow x = r\cos \theta - - - - \left( 1 \right)$
$\Rightarrow y = r\sin \theta$
Also, we know that $sec\theta = \dfrac{1}{{\cos \theta }}$
Now it is given in the question that $r = 7sec\theta \;$.
Now substitute the value $sec\theta = \dfrac{1}{{\cos \theta }}$ in above equation, we get
$\Rightarrow r = \dfrac{7}{{\cos \theta }}$
Now, on cross-multiplication
$\Rightarrow r\cos \theta = 7$
Now, from equation $1$ , we get
$\Rightarrow x = 7$
From the above equation it is clear that it is a vertical line, parallel to the y-axis.
Therefore, equation $r = 7sec\theta \;$ can be written in the rectangular form as $x = 7\,\,$.
So, the correct answer is “ $x = 7\,\,$”.

Note : Polar equations can be converted into a rectangular form by using the relation between their coordinates. Whenever we talk about polar form, we must think of the inclination of that point with the positive x-axis. We can convert the x coordinate into polar form as $x\, = \,r\cos \theta$ and y coordinate as $y = r\sin \theta$. $\theta$ gives the inclination of a line with positive direction of the x axis.