# Rewriting a weird equation

#### Illvoices

##### Junior Member
Hello I've got a question for you guys and I start by asking where is the science in writing the equation of a circle with (h,k) and radius r.
(x-h)2+(y+k)2=r2

#### Dr.Peterson

##### Elite Member
Huh? "Where is the science?" Please explain.

That form of the equation of a circle is just an expression of the distance formula.

#### Otis

##### Senior Member
… I've got a question … I start by asking where is the science
Is that the question you've got, or are you starting with a second question?

I'm not sure what you're thinking about science. Can you post an example?

(x - h)2 + (y + k)2 = r2
That plus sign is a typo.

(x - h)2 + (y - k)2 = r2

The circle's center is the point (h, k).

#### Illvoices

##### Junior Member
So this is a distance formula. Yea Khan is using this one to prep for the SAT's, I just didn't find it real since its the first time they use such a formula it's a little bit intimidating. Oh and yes there is a typo mr. Cat it is a negative sign I just realized this. I didn't know these facts so thanks I will be posting an a example when I'm faced with a second problem.

#### HallsofIvy

##### Elite Member
"Mathematics" is NOT "science"! It makes no sense to ask "what is the science" of a given equation. Of course many things in science (as well as other fields) can be represented in mathematics so as equations but exactly what that equation is depends upon how you are "modeling" the science.

If you are asking "Why is that the equation of a circle", then you have to specify that you are representing x and y in a "Cartesian coordinate system" where the x and y axes are at right angles. Given a point (x, y), the line segment from (0, 0) to (x, 0) has length x and the line segment from (x, 0) to (x, y) has length y. By the "Pythagorean theorem" (still "mathematics", not "science"!) the distance from (0, 0) to (x, y) has length $$\displaystyle \sqrt{x^2+ y^2}$$.

All the points on a circle with center at (0, 0) and radius r have distanced r from (0,0) (pretty much the definition of "circle") so $$\displaystyle r= \sqrt{x^2+ y^2}$$ and. squaring both sides, $$\displaystyle r^2= x^2+ y^2$$.