Advanced Search

Question: Rewriting general formula

Question:Rewriting general formula

janhardo 335 Maple
spam simplification conic duplicate-question
July 13 2020
0

Some rewriting for a general equation , how to do ?


 

Transformation of eq 1 in wanted eq 2 form

A*x^2+B*x*y+C*y^2+D*x+E*y+F = 0 (general form 2)

After complete square for x and y and some rearrangment (general form 2) ,  i got this equation eq1  

 

 

> 

(A*(x + (B*y + D)/(2*A))^2 + C*(y + (B*x + E)/(2*C))^2)/(-C*y^2 - E*y - 2*F + (B*y + D)^2/(4*A) - A*x^2 - D*x + (B*x + E)^2/(4*C))=1;

(A*(x+(1/2)*(B*y+D)/A)^2+C*(y+(1/2)*(B*x+E)/C)^2)/(-C*y^2-E*y-2*F+(1/4)*(B*y+D)^2/A-A*x^2-D*x+(1/4)*(B*x+E)^2/C) = 1

 (1)
> 

eg1:= (-4*A^2*C*x^2 + (-4*y^2*C^2 + (-8*B*x*y - 4*D*x - 4*E*y)*C - (B*x + E)^2)*A - (B*y + D)^2*C)/(4*A^2*C*x^2 + (4*y^2*C^2 + (4*D*x + 4*E*y + 8*F)*C - (B*x + E)^2)*A - (B*y + D)^2*C) = 1;

(-4*A^2*C*x^2+(-4*y^2*C^2+(-8*B*x*y-4*D*x-4*E*y)*C-(B*x+E)^2)*A-(B*y+D)^2*C)/(4*A^2*C*x^2+(4*y^2*C^2+(4*D*x+4*E*y+8*F)*C-(B*x+E)^2)*A-(B*y+D)^2*C) = 1

 (2)
> 

 

(1) wanted form of eg1
"A x^2+B x y+C y^2+D x+E y+F=0*("general form 2))

for M ≠ 0

 

(x+(B*y+D)/(2*A))^2/X+  (y+(B*x+E)/(2*C))^2/Y = 1

 

Note : what X and Y could be ? :   M/A and M/B  ..

So it must be possible to transform eq1  in this form  above ?

> 

 

Some background info

If B= 0 in (general form 2) you get (general form 1)  

 

(2) In book for M ≠ 0 as example we get this form
 Ax^2+Cy^2+Dx+Ey+F = 0(general form 1 )

(x+C/(2*A))^2*A/M+  (y+D/(2*B))^2*B/M = 1

> 

 


 

Download vraag1-herleidingconics2.mw

 

 
Please Wait...