if a curve is symmetrical about each of the coordinate axis than prove that it is also symmetrical about the - Maths - Conic Sections

Question

if a curve is symmetrical about each of the coordinate axis than
prove that it is also symmetrical about the origin What can you say
about the converse

Neha Sethi · Accepted Answer

Dear student
Let Pα,β be  a point on the curve C,then Fα,β=0    
1Now the curve C is symmterical about x axis iff the reflection of Pin the x axis ,ie the point P'α,-β lies on C i
e   iff Fα,-β=0  
2From 1 and 2, it follows that C is symmterical about x axis iffFx,y=Fx,-ySimilalrly C is symmterical about y axis iffFx,y=F-x,yLet Pα,β be  apoint on curve C then Fα,β=0Now the curve is symmetrical about origin iff the reflection of P in the originie  P'-α,-β lies on Cie iff F-α,-β=0From 1 and 2 it follows that  C is symmterical about origin ifFx,y=F-x,-yAbout converse we cannot say for example xy-1=0
Regards

if a curve is symmetrical about each of the coordinate axis than prove that it is also symmetrical about the origin.What can you say about the converse