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

  • Thread starter zorro
  • Start date
1,380
0
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

Science Advisor
Homework Helper
41,709
876
Any conic section can be written in the form [itex]f(x, y)= A(x- x_0)^2+ B(y- y_0)^2= C[/itex] for some number A and B, in some coordinate system (with coordinate axes parallel to the axes of symmetry of the conic section), and [itex](x_0, y_0)[/itex] as center in that coordinate system.

In this case, [itex]f_x= 2A(x- x_0)= 0[/itex] and [itex]f_y= 2B(y- y_0)= 0[/itex] so that [itex]x= x_0[/itex] and [itex]y= y_0[/itex]. 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.
 
1,380
0
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

Science Advisor
Homework Helper
41,709
876
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 [itex]\theta[/itex].

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.
 
1,380
0
Thanks!
I got it.
 

Related Threads for: Method to find the centre of a conic section from its equation

Replies
3
Views
4K
Replies
11
Views
924
Replies
18
Views
1K
  • Posted
Replies
5
Views
3K
  • Posted
Replies
4
Views
2K
  • Posted
Replies
9
Views
4K
  • Posted
Replies
1
Views
2K
Replies
1
Views
2K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Hot Threads

Top