Machinery's Handbook, 31st Edition
HYPERBOLA
59
9 x 2 4 y 2 – 36 x – 8 y + – 4 9 x 2 36 x – ( =
2 8 y –
) 4 y ( –
=
4
) )
9 x 2 4 x – +4 (
2 2 y – +1
) 4 y ( –
= 36
2
9 x – 2 ( ) 36 ---------- 4 y – 1 ( ) 2 36 – ----------- ----
x – 2 ( ) 2 2 ---------- y – 1 ( ) 2 3 2 2
=
– ---------- 1 =
Comparing the results above with the form x h – ( ) 2 a 2 ----------
2
y k – ( ) 2 – --------- = 1 and calculating
b
2
2
+
b
a
2
2
eccentricity from e
a = ----------- and distance c from c
gives
a = +
b
13 2 = -----
h = 2
k = 1 ,
a = 2
b = 3
c = 13
e
,
,
,
,
Complex Numbers Imaginary Number.— The square root of a negative number cannot be expressed with real numbers, since any negative number multiplied by itself is positive. But technical mathematics often relies on computation involving the square root of –1. For this, imagi- nary number i is defined as follows: i = − − = − ( ) 1 1 1 2 , so Imaginary numbers are not real numbers; they belong to the set of complex numbers . An example of an equation that cannot be solved with real numbers is x 2 + 1 = 0. Re arranging gives x 2 = –1, and taking the square root of both sides gives x i = ± − = 1 ± . (Note: The letter j is also used to represent the imaginary number − 1 .) Forms of a Complex Number.— Complex numbers can be expressed in several forms, all of which are based on the complex coordinate system, as seen in Fig. 1 and Fig. 2. Operations on complex numbers: Complex numbers are added and subtracted much like real numbers, but with real parts added to real parts and imaginary to imaginary:
( a + bi ) + ( c + di ) = ( a + c ) + ( bi + di ) = ( a + c ) + ( b + d ) i ( a + bi ) – ( c + di ) = ( a – c ) + ( bi – di ) = ( a – c ) + ( b – d ) i
where coefficients a, b, c, and d are real numbers. Example 1: (3 + 4 i ) + (2 – i ) = (3 + 2) + (4 i – i ) = 5 + 3 i Complex numbers are multiplied as binomials are, by FOIL:
( a + bi )( c + di ) = ac + adi + bci + bdi 2 = ac + ( ad + bc ) i + bd (–1) = ac + ( ad + bc ) i – bd Example 2: (1 + 2 i )(5 – 7 i ) = 5 – 7 i + 10 i –14 i 2 = 5 + 3 i – (14)(–1) = 5 +3 i + 14 = 19 + 3 i Standard (rectangular) form of a complex number: A complex number z has a real part and an imaginary part. Its standard form is z = a + bi , where a is the real part and bi the imaginary part. Fig. 1 shows how the complex plane is similar to the real plane (see page 42 in Analytic Geometry ), except here only the horizontal axis, x , is real, whereas the vertical axis, yi , is imaginary. Fig. 2 shows examples of complex numbers in rectan- gular form.
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