When my professor was going through coninc sections of the real plane he didn't really mention why such a curve has no real points iff the determinant of the matrix associated to the conic, A, without the first row and column is positive (this I get it, of course, it has to be an ellipse) AND a(1,1)*det(A)>0.
The second part is the one that messes me up: I've been trying to understand why it works this way for the last three days, unsuccessfully. Could someone show me a proof? My books don't even talk about it and I haven't found much on the web so far.
Thanks in advance!
By proof I mean how do I prove that parabolas, hyperbolas and ellipses are intersections between a plane and cone?
I understand the importance of circles, parabolas, eclipses, etc. But why even mention they're from a cone? I tried searching up a reason but couldn't find anything (just stuff on why the SECTIONS are important, which I already knew).
Just feel like it's irrelavent to even mention they come from a cone. (almost like it's just trivia). Anyone can enlighten me?
I’m an undergrad who has to take 2 semesters of calc, and I’m struggling to figure out why those conics are so important
What are degenerate conic sections? and how are they generated/formed?
i lack the latex skills to format equations properly, so i'll link a picture to the question from the textbook, as well as the steps and solution i managed to get, right here
basically, i reach a weird form: (x-h) +4 (y-k)= 1 which i attempt to standardise into: (x-h)/1 + (y-k)/¼ = 1 which looks very unsettling and very likely wrong . help me?
can someone please help me do these word problems because i really have a hard time doing word problems. here's the word problems:
1.the towers supporting the cable suspension bridge are 1200m apart and 170m above the bridge it supports. SUppose the cable hands, following the shape of a parabla, with its lowest point 20m above the bridge How high is the cable 120m away from a tower?
2.A whisper gallery has a semielliptical ceiling that is 9m high and 30m long how high is the ceiling above the two foci?