Machinery's Handbook, 31st Edition
58
HYPERBOLA
Thus, k = 3 ⁄
2 , h = - 1 and p =
1 ⁄ 2 . Focus F is located at point ( h + p , k ) = ( 1 ⁄ 2 , 3 ⁄
2 ); the
directrix is located at x = h - p = - 1 - 1 ⁄
2 = -
3 ⁄ 2 ; the parabolic axis is the horizontal line 2 ); and the latus rectum lies on the
y = 3 ⁄ 2 ; the vertex V ( h , k ) is located at point ( - 1, 3 ⁄ line x = h + p = - 1 ⁄ 2 ) = 2. Hyperbola.— Referring to the figure on the left, below, a hyperbola is the set of all points such that | d 1 – d 2 | is constant. That is, the difference between the distances from any point ( x, y ) to the foci, marked F 1 (– c , 0) and F 2 ( c , 0) does not change. The distance between the vertices , V 1 (–a , 0) and V 2 ( a , 0) (the turning points of the hyperbola) is 2 a . Therefore, | d 1 – d 2 | = 2 a for any two points on the hyperbola. 2 . Its length is 4 | - 1 ⁄ 2 | = 4( 1 ⁄ The figure on the right shows more detail. The slopes of the asymptotes (lines of approach) relate to the transverse and conjugate axis lengths, 2 a and 2 b . The cen- ter of a hyperbola is the point of intersection of the asymptotes. In the figure, the cen - ter is shown as the origin, (0, 0). The standard form of the hyperbola, as derived from the foci with center at the origin, is x a y b 2 2 2 2 1 − = . For any center ( h, k ), the equation is ( ) ( ) x h a y k b − − − = 2 2 2 2 1 .
The general form of the hyperbola is given by Ax 2 + By 2 + Cx + Dy + E = 0, where AB < 0 and AB ≠ 0.
2
2
+
b
a
Also, the eccentricity of a hyperbola, e
a = ----------- , is always less than 1.
The distance 2 c between the two foci is given by the relation: 2 c . Example: Determine the values of h , k , a , b , c , and e of the hyperbola general form: 2 a 2 b 2 + =
2
2
9 x
–
4 y
36 x – 8 y + – 4 = 0
Solution: Convert the hyperbola equation into the standard form (see Solving by Com- pleting the Square on page 34):
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