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The Hyperbola

The hyperbola is the set of points P in the plane such that the absolute value of the difference of the distances from P to two fixed points F1 & F2 is a constant.

| F1P - F2P | = k

The fixed points are the foci of the of the hyperbola. Using the image of the stacked cones again, the shape of the parabola is more apparent:

The general equation for the hyperbola is , when it is in the standard position (like above). Notice the similarties to the equation of the ellipse.

The standard equation for the hyperbola is

The key points for the hyperbola in standard position are:

Information for the hyperbola:

· Length of transverse axis is 2a

· Length of Conjugate axis is 2b

· Vertices are at (-a,0) & (a,0) and the co-vertices are at (0,b) and (0, -b)

o Transverse along y axis then Vertices (0, ± a)  , co-vertices at ( ± b, 0)

· The foci are at (-c,v0) & (c,v0) and are on the transverse axis

o Transverse along y axis Foci at (0, ± c )

· The asymptotes are at ;

· a 2 + b 2 = c 2

For the ellipse with the transverse axis on the y axis, the major points are on the y, and x axis coordinates are zero (flipped points)

The equation is

 

Example:

Given , find the loci and the asymptotes:

This is in the form of an ellipse with the transverse axis along the y axis. The vertices will be at (0,-6) and (0,6).the foci are at

Hence the foci are at  (0, ) & (0,).

The asymptotes are at

For hyperbolas not centered at the origin:

The standard form of the equation of a hyperbola centered at (h, k), with the transverse axis parallel to the x-axis and the conjugate axis parallel to the y-axis, is

The length of the transverse axis parallel to the x-axis is 2a.

The length of the conjugate axis parallel to the y-axis is 2b.

The vertices are V 1 (h - a, k) and V 2 (h + a, k).

The co-vertices are (h, k - b) and (h, k + b).

The foci are F 1 (h - c, k) and F 2 (h + c, k), which are on the transverse axis.

a 2 + b 2 = c 2

The standard form of the equation of a hyperbola centered at (h, k), with the transverse axis parallel to the y-axis and the conjugate axis parallel to the x-axis, is

The length of the transverse axis parallel to the y-axis is 2a.

The length of the conjugate axis parallel to the x-axis is 2b.

The vertices are V 1 (h, k - a) and V 2 (h, k + a).

The co-vertices are (h - b, k) and (h + b, k).

The foci are F 1 (h - c, k) and F 2 (h + c, k), which are on the transverse axis.

a 2 + b 2 = c 2

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