Hyperbola

A hyperbola is the locus of the points on the plane whose distance difference r '- r, at two fixed points F and F', called foci, is constant and equal to 2a, the latter being the length of the actual axis AB of the hyperbola.

Hyperbola

Equation of a hyperbola

The equation x2 / to2 - Y2 / b2 = 1 is called the canonical equation of a hyperbola.

Hyperbola 2

  • Spotlights: They are the fixed points F and F '.
  • Main or real axis: is the line that passes through the foci.
  • Secondary or imaginary axis: is the perpendicular bisector of the segment FF '.
  • Center: is the point of intersection of the axes.
  • Vertices: Points A and A 'are the intersection points of the hyperbola with the focal axis.
  • Focal distance: is the FF segment of length 2c.
  • Major axis: is the segment of length AA 2a.
  • Minor axis: is the segment (BB ') ̅ of length 2b.
  • Asymptotes: are the lines of equations: lines: y = bx / a; y = -bx / a
  • Semi major axis = to
  • Semi minor axis = b The three semi-axes satisfy   c2 = to2 + b2
  • Semi focal axis = c
  • Eccentricity e: Determines the shape of the curve, in the sense of whether it is more rounded or if it approaches a segment.

Example 1: Determine the foci, vertices, and asymptotes of the hyperbola: x2 / 9 - and2 / 16 = 1

From the equation we know:

 a = 3; b = 4

 Now as c2 = to2 + b2 so:

c2 = b2 + a2

c = √ (16 + 9)

c = √25 = 5

Thus:

 Spotlights: F (5.0) and F '(-5.0); Vertices: A (3,0) and F '(-2,0); Asymptotes: y = 4 / 3x; y = - 4 / 3x