Question

How do you convert $2\sin \theta - 3\cos \theta = r$ to rectangular form?

Answer 1

Here we need to convert the given polar form into the rectangular form. We will write the polar coordinates and the rectangular coordinates. Then we will use the relation between the polar coordinates and the rectangular coordinates. Then we will substitute all these values in the given polar form and from there, we will get the required rectangular form.

Complete step by step answer:
Here we need to convert the given polar form into the rectangular form and the given polar form is $2\sin \theta - 3\cos \theta = r$.
We know if $x$ and $y$ are the rectangular coordinates and $r$ and $\theta$ are the polar coordinates then
$x = r\cos \theta$ …………… $\left( 1 \right)$
$y = r\sin \theta$ …………. $\left( 2 \right)$
We also know that
${x^2} + {y^2} = {r^2}$ ……………… $\left( 3 \right)$
Now, we will multiply both sides of the equation $2\sin \theta - 3\cos \theta = r$ by the term $r$.
$\Rightarrow$ $2r\sin \theta - 3r\cos \theta = {r^2}$
Now, we will substitute the values from equation 1, equation 2, and equation 3 in the above equation, we get.
$\Rightarrow 2y - 3x = {x^2} + {y^2}$
On further simplifying the terms, we get
$\Rightarrow {x^2} + {y^2} - 2y + 3x = 0$
Hence, this is the required conversion of the given polar form to the rectangular form.

Note: Here we have converted the given polar form into the rectangular form. We have used the relation between the polar and the rectangular coordinates to convert the given polar form into the rectangular form. So we can also use these relations between the polar and the rectangular coordinates to convert the rectangular coordinates into the polar coordinates. The polar coordinate system is defined as the two-dimensional coordinate system in which each point on a plane is determined by a distance of that point from a reference or the center point and an angle from a reference direction.