Where does the general form of conics comes from?

[surreal narwhal]

I know many books just define this, and say this is the general equation for conics, but how does this relates to a conic sections? Like, how do we get this equation from the geometric idea?

[wide wing]

1. Ask your question and show the work you've done so far. If you've posted a screenshot of a question, specify which part you need help with.
2. Wait patiently for a helper to come along.
3. Once someone helps you, say thank you and close the thread with:

   +close
   

4. Feel free to nominate the person for helper of the week in #helper-nominations
5. Do not ping the mods, unless someone is breaking the rules.
6. If you're happy with the help you got here, and the server overall, you can contribute financially as well:

[warm pivot]

Geometric idk, but you should know the general equations of all the possible conics I assume
Just show that they all fall under that crazy equation (ok it’s understandable)

[surreal narwhal]

That part makes sense I mean

I thought of it as just a kind of label that encapsulates all possible equations

But I thought there was more meaning behind it

[warm pivot]

There probably is an intuitive sense tbh

[unborn heart]

Check out Apollonius, Pappus and Descartes

#1253543911759151154 message

[wind mica]

it's a second degree equation

Every 2nd degree equation is a conic just factor it

here's a proof from my book

[surreal narwhal]

well thank you my brother

[untold idol]

@surreal narwhal has given 1 rep to @unborn heart

[surreal narwhal]

AMAZING

thank you sir

you guys saved a soul today

thanks @wind mica

oh well

[random kraken]

Take a cone z^2 = x^2 + y^2 and intersect it with an arbitrary plane ax + by + cz + d = 0.
You can express z from the equation of the plane and substitute it into the equation of the cone. Probably better to multiply both parts by c^2 first, though:
c^2 z^2 = c^2 (x^2 + y^2)
So, we get:
(ax + by + d)^2 = c^2 (x^2 + y^2)
And you can continue from here.

[surreal narwhal]

by far the most simple explanation and now i really get it

https://tenor.com/view/always-sunny-frank-oh-my-god-i-get-it-gif-22328165

thank you sir