Question

How do you write the equation \[6x - 8y = 21\] in polar form?

Answer 1

Hint: Here in the above question we need to write the equation in polar form. So we will solve it by finding the conversion formula between the Cartesian coordinate system and the polar coordinate system, thus replacing \[x\] and \[y\] with its polar form. By the use of the information that polar form consists of and we can write the equation in polar form.
Formula: The basic conversion formula used for converting Cartesian to polar coordinate system is
 \[
   \Rightarrow x = r\cos \theta \\
   \Rightarrow y = r\cos \theta \\
   \Rightarrow r = \sqrt {{x^2} + {y^2}} \\
   \Rightarrow \tan \theta = \dfrac{y}{x} \;
 \]
So for the above problem by the use of the first two formulas we will replace the variables in the given equation with the polar coordinates and later on simplify the equation forgetting the desired result.

Complete step by step solution:
The equation given that we need to convert to polar form is \[6x - 8y = 21\]
By the analysis of the equation we can find that there are two variables \[x\] and y in the given equation. As we are aware that \[x\] and y belongs to the Cartesian system of coordinates. Now for the conversion of Cartesian to polar coordinate system remember the sets of formula is
 \[
   \Rightarrow x = r\cos \theta \\
   \Rightarrow y = r\cos \theta \;
 \]
Now we will substitute the value in the equation and obtain
 \[
   \Rightarrow 6(r\cos \theta ) - 8(r\sin \theta ) = 21 \\
   \Rightarrow 2r(3\cos \theta - 4\sin \theta ) = 21 \\
   \Rightarrow r = \dfrac{{21}}{{2(3\cos \theta - 4\sin \theta )}} \;
 \]
Therefore it is concluded that polar form of the equation is \[r = \dfrac{{21}}{{2(3\cos \theta - 4\sin \theta )}}\]
So, the correct answer is “ \[r = \dfrac{{21}}{{2(3\cos \theta - 4\sin \theta )}}\] ”.

Note: Polar form consists of and as its coordinates and \[r\] stands for distance from the origin \[\theta \] stands for the angle of the particular coordinate in the antilock wise direction with positive \[x\] axis .
Notes: In order to convert the equation of such types we need to remember the basic conversion formulae otherwise it will not be easier to solve it. While converting we need to take care of the signs, the degrees of the term. Keep in mind that we need to simplify the equation as far as possible in order to get the desired results.