I need to learn how to find the properties (foci, vertex, etc) of a conic given its equation, I know

mistergoneo7 2022-07-02 Answered
I need to learn how to find the properties (foci, vertex, etc) of a conic given its equation, I know how to do it when the coeficient of x y = 0 , but I have troubles in the other case. Can someone recommend me a book, website o document where I can learn how to find the properties of a conic section given its equation please?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Gornil2
Answered 2022-07-03 Author has 20 answers
Step 1
Given the equation of the conic in the form
A x 2 + B x y + C y 2 + D x + E y + F = 0
The term containing xy can be eliminated using a rotation. Define the change of variables as follows
x = x cos θ y sin θ
y = x sin θ + y cos θ
A point (x, y) will have coordinates ( x , y ) in a coordinate frame that is rotated by θ counter-clockwise with respect to the original frame.
Plug these in,
A ( x 2 cos 2 θ + y 2 sin 2 θ x y sin 2 θ ) + B ( 1 2 ( x 2 y 2 ) sin 2 θ + x y cos 2 θ ) + C ( x 2 sin 2 θ + y 2 cos 2 θ + x y sin 2 θ ) + D ( x cos θ y sin θ ) + E ( x sin θ + y cos θ ) + F = 0
Combining like terms,
x 2 ( A cos 2 θ + C sin 2 θ + 1 2 B sin 2 θ ) + y 2 ( A sin 2 θ + C cos 2 θ 1 2 sin 2 θ ) + x y ( ( C A ) sin 2 θ + B cos 2 θ ) + D ( x cos θ y sin θ ) + E ( x sin θ + y cos θ ) + F = 0
Now we want to eliminate the x y term, so we have to choose θ such that
( C A ) sin 2 θ + B cos 2 θ = 0
that is,
tan 2 θ = B A C
Now the equation is
A x 2 + C y 2 + D x + E y + F = 0
where
A = A cos 2 θ + C sin 2 θ + 1 2 B sin 2 θ
C = A sin 2 θ + C cos 2 θ 1 2 B sin 2 θ
D = D cos θ + E sin θ
E = D sin θ + E cos θ
F = F
Since rotation preserves lengths, you can find the axes, foci, vertices, and then rotate everything by the angle θ to obtain them in the original coordinate frame.
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

New questions