Question

How do you convert `3y= 2x^2-2xy-7y^2` into a polar equation?

Answer 1

Substitute `rcos(theta)` for `x` and `rsin(theta)` for y.
Write `r` as a function of `theta`

Explanation:

Given: `3y= 2x^2-2xy-7y^2`

Here is the graph of the Cartesian equation:

Substitute `rcos(theta)` for `x` and `rsin(theta)` for y.

#3rsin(theta)= 2(rcos(theta))^2-2(rcos(theta))(rsin(theta))-7(rsin(theta))^2

Write `r` as a function of `theta`

`3sin(theta)r = (2cos^2(theta) -2cos(theta)sin(theta)-7sin^2(theta))r^2`

Please observe that we can safely divide by r, because that will only eliminate the trivial root `r = 0`:

`3sin(theta) = (2cos^2(theta) -2cos(theta)sin(theta)-7sin^2(theta))r`

Divide by the coefficient in front of r:

`r = (3sin(theta))/(2cos^2(theta) -2cos(theta)sin(theta)-7sin^2(theta))`

Here is the graph the polar equation.

This proves that the conversion is done properly.