Definition:
A hyperbola is a type of a smooth curve, lying in a plane, defined by its geometric properties or by equations for which it is the solution set. This is represented in the following graph.
1. The linear eccentricity (c) is the distance between the center and the focus (or one of the two foci).
2. The latus rectum (2γ) is the chord passing through the focus (or one of the two foci).
3. The Vertex is a Point in the axis where hyperbola intersect.
4. The Asymptotic Lines is a line that comes along a curve but never touches the curve.
5. The Foci is any fixed point which is located inside the curve along the axis.
Formula:
Eccentricity (c) = √(a2 + b2)
Foci = (0,c) & (0,c)
Vertices = (0,a) & (0,a)
Asymptotic lines = (b/a)x & (b/a)x
Latus rectum = 2b^{2}/a
Example:
Find the Eccentricity, Foci, vertices, Asymptotic lines and Latus rectum of hyperbola with a point (3,2)?
Given:
a = 3
b = 2
To Find,
Eccentricity, Foci, Vertices, Asymptotic lines and Latus rectum
Solution:
Step 1 :
Eccentricity (c) = √(a2 + b2)
c = √(32 + 22)
c = √(9+4) = √13 = 3.6
Step 2 :
Foci = (0,c) & (0,c)
Foci = (0,3.6) & (0,3.6)
Step 3 :
Vertices = (0,a) & (0,a)
Vertices = (0,3) & (0,3)
Step 4 :
Asymptotic lines = (b/a)x & (b/a)x
Asymptotic lines = (2/3)x & (2/3)x
Asymptotic lines = (0.67)x & (0.67)x
Step 5 :
Latus rectum = 2b^{2}/a
Latus rectum = 2*22/3
Latus rectum = 8/3 = 2.67
