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 any one explain me?

HallsofIvy

Homework Helper
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

Homework Helper
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.

• Ravi tej

Thanks!
I got it.

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