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.

**Equation of a hyperbola**

The equation x^{2 }/ to^{2} - Y^{2} / b^{2} = 1 is called the **canonical equation of a hyperbola**.

**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**c**^{2 }= to^{2 }+ b^{2}**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: x^{2 }/ 9 - and^{2 }/ 16 = 1

From the equation we know:

a = 3; b = 4

Now as c^{2 }= to^{2 }+ b^{2} so:

c^{2 }= b^{2 }+ a^{2}

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