Method to find the centre of a conic section from its equation

[zorro]

In the second degree equation of a conic section (ellipse/hyperbola), I have seen many books following this method to find out the centre of the conic section-

1) Differentiate the equation w.r.t x treating y as constant
2) Differentiate the equation w.r.t y treating x as constant.
3) Solve the above two equations to find out the centre of the curve

I searched many books but did not find the theory behind it.
Can anyone explain me?

 

[HallsofIvy]

Any conic section can be written in the form $f(x, y)= A(x- x_0)^2+ B(y- y_0)^2= C$ for some number A and B, in some coordinate system (with coordinate axes parallel to the axes of symmetry of the conic section), and $(x_0, y_0)$ as center in that coordinate system.

In this case, $f_x= 2A(x- x_0)= 0$ and $f_y= 2B(y- y_0)= 0$ so that $x= x_0$ and $y= y_0$. For the general equation you need that any coordinate system can be transformed into this coordinate system by rotations and translations which transform linear equations into linear equations.

 

[zorro]

which transform linear equations into linear equations.

I did not get this.
One more thing, by the process of differentiation, are we changing the co-ordinate system?

 

[HallsofIvy]

I did not get this.

Do you understand what I mean by "rotations" and "translations"? What happens, say, to the line y= mx if you translate it by adding a to x and adding b to y? What happens if you rotate around the origin by an angle $\theta$.

One more thing, by the process of differentiation, are we changing the co-ordinate system?

Of course not. In order to be able to differentiate with respect to "x" and "y", we must have variables "x" and "y" which means a specific coordinate system.

 

[zorro]

Thanks!
I got it.