fresh_42 said:The symmetry comes from the absolute value. One branch for each sign of ##||PF_2|| - ||PF_1||##. For ##a=0## one gets the degenerate hyperbola, a straight, the height of a double pyramide. Maybe I didn't catch your point.
A hyperbola is a type of conic section, along with circles, ellipses, and parabolas. It is a curve that is formed by the intersection of a plane with a double cone.
The standard equation of a hyperbola is (x-h)^2/a^2 - (y-k)^2/b^2 = 1, where (h,k) is the center of the hyperbola and a and b are the distances from the center to the vertices along the x and y axes, respectively.
While both are conic sections, an ellipse is a closed curve with two foci, while a hyperbola is an open curve with two branches that extend infinitely. Additionally, the sum of the distances from any point on an ellipse to the two foci is constant, while the difference of distances from any point on a hyperbola to the two foci is constant.
Hyperbolas have practical applications in fields such as engineering, physics, and astronomy. They can be used to model the orbits of planets and comets, as well as the paths of radio waves and satellite signals. They also have applications in optics, such as in the design of satellite dishes and telescopes.
To graph a hyperbola, you can plot the center point and the vertices, which can be found using the equation of the hyperbola. Then, draw the asymptotes, which are lines that pass through the center and intersect the vertices. Finally, sketch the two branches of the hyperbola, using the asymptotes as guides for their shape and direction.