To be honest, it should be called `Polar Coordinate System`

> In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.

- The `reference point` (analogous to the origin of a Cartesian coordinate system, the center point) is called the `pole`
- The `ray` from the pole in the reference direction is the `polar axis`

You can represent any point in this system with `(distance, angle)`, for example, `(3, 60 degree)`

or generally,

: rho

: theta

### Converting between polar and Cartesian coordinates

# Question and Answers

#### In polar coordinate system, a circle equation is
, a line equation is
, what's the max distance for a point at that circle to the line?

> the answer is 6

##### First, we need to convert all information from `polar coordinate` to `Cartesian coordinate`:

###### 1. circle

so now we know the center point of that circle is `(0, 4)`, radius of the circle is `4`

###### 2. line

we know
is nothing but a line of 60 degree angle.

And we also know a general line could be represented as

And

so:

##### How to get the max distance?

Inner a circle, `diameter` is the longest line segment you can get.

> A special line drawn inside a circle is called a chord. A chord can be of different lengths. The longest length of the chord is called a diameter. Two times the radius makes a diameter.

And for the distance between a point(
) and a line(
), there's a formula for it:

So first, we need to choose a line as close a diameter as possible.

For some reason, people already known `the distance between a line and circle center` + `the radius of that circle` = `the max distance for a point at that circle to the line`.

so:

Thank you for reading, I love the feeling of doing this.

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