- #1

PainterGuy

- 940

- 70

- TL;DR Summary
- Why this equation, x²+y²+2ax+2by+c=0, is called 'normal form of the conic section equation'.

Hi,

The following is called normal form of the conic section equation:

x²+y²+2ax+2by+c=0

A circle is one of the conic sections when considered as a special of ellipse. I'm confused as to why the the given equation is called "normal form of the conic section equation" when, in my opinion, the equation only describes a circle, a point which could be said to be a special case of circle, imaginary locus or imaginary circle; as I understand it the equation has nothing to do with the conic sections other than the circle which is a special case. Could it represent any other conic section?

The shown below is equation in its context:

I have tried below to relate the "normal form of the conic section equation" to general equation of a circle with radius R and center at (x₀, y₀).

Thank you for your help and time!

The following is called normal form of the conic section equation:

x²+y²+2ax+2by+c=0

A circle is one of the conic sections when considered as a special of ellipse. I'm confused as to why the the given equation is called "normal form of the conic section equation" when, in my opinion, the equation only describes a circle, a point which could be said to be a special case of circle, imaginary locus or imaginary circle; as I understand it the equation has nothing to do with the conic sections other than the circle which is a special case. Could it represent any other conic section?

The shown below is equation in its context:

I have tried below to relate the "normal form of the conic section equation" to general equation of a circle with radius R and center at (x₀, y₀).

Thank you for your help and time!