#Co-ordinate geometry help
signal pollen
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summer cargo
Do you remember the general form of the equation of the circle?
signal pollen
yeah
summer cargo
What is it?
summer cargo
Well, that's only if it's centered at the origin.
signal pollen
(x-h)^2 + (y-p)^2 = r^2
summer cargo
Right! Now, let's transform it a bit.
Open both brackets and collect similar terms (as in, quadratic terms, linear terms and constant term).
summer cargo
As I said, collect similar terms.
There are five kinds of terms here: x^2, y^2, x, y and constant.
summer cargo
Well, more or less.
Now, notice how this can be written in the following form:
x^2 + y^2 + Ax + By + C = 0
Can you see why that's true?
signal pollen
Yes
summer cargo
Nice.
Now, notice how if you substitute coordinates of a point into that equation, you'll get a linear equation in A, B, C.
So, if you substitute three different pairs of coordinates, you get a system of equations.
signal pollen
i get 3 simultaneous equations?
summer cargo
Yeah. A system of three linear equations.
So, you can find the coefficients from it.
summer cargo
Well, yeah, solving linear systems isn't very exciting. But oh well, what can we do 😅
summer cargo
No.
Work with the equation we've obtained: x^2 + y^2 + Ax + By + C = 0.
Otherwise the equations you'll get won't be linear, and it's harder to solve those.
signal pollen
ohhh yeah