How do I find the polar equation for $x^{2} + y^{2} = 7 y$ $x^2+y^2=7y$ ?

Question

14162 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2014-09-21T00:00:21

To solve this problem we need to convert all of the $x$ $x$ 's and $y$ $y$ 's to $r$ $r$ 's and $θ$ $theta$ 's.

Before we look at the specifics of this problem please take a look at the relationships between the polar and rectangular coordinate systems.

Understanding these relationships are critical to better understanding how to solve these types of problems.

$x = r cos (θ)$ $x=rcos(theta)$

$y = r sin (θ)$ $y=rsin(theta)$

$\frac{y}{x} = tan (θ)$ $y/x=tan(theta)$

${tan}^{1} (\frac{y}{x}) = θ$ $tan^1(y/x)=theta$

$x^{2} + y^{2} = r^{2}$ $x^2+y^2=r^2$

To solve this type of problem one of the first things to look for is a way to make substitutions.

$x^{2} + y^{2} = 7 y$ $x^2+y^2=7y$

Substitute in $r^{2}$ $r^2$ for $x^{2} + y^{2}$ $x^2+y^2$

$r^{2} = 7 y$ $r^2=7y$

Substitute in $r sin (θ)$ $rsin(theta)$ for $y$ $y$

$r^{2} = 7 r sin (θ)$ $r^2=7rsin(theta)$

$\frac{r^{2}}{r} = \frac{7 r sin (θ)}{r}$ $(r^2)/r=(7rsin(theta))/r$

$r = 7 sin (θ) \to$ $r=7sin(theta) ->$ is the polar form of $\to x^{2} + y^{2} = 7 y$ $-> x^2+y^2=7y$

How do I find the polar equation for x^2+y^2=7yx2+y2=7yx^2+y^2=7y?