If a a 2 b 2 z 1 a a 2 b 2 z 2 we have z 1 2 z 2 2 1

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[If a + a 2 - b 2 = z 1 , a - a 2 - b 2 = z 2 , we have | z 1 | 2 + | z 2 | 2 = 1 2 | z 1 + z 2 | 2 + 1 2 | z 1 - z 2 | 2 = 2 | a | 2 + 2 | a 2 - b 2 | , and so ( | z 1 | + | z 2 | ) 2 = 2 {| a | 2 + | a 2 - b 2 | + | b | 2 } = | a + b | 2 + | a - b | 2 + 2 | a 2 - b 2 | . Another way of stating the result is: if z 1 and z 2 are the roots of αz 2 + 2 βz + γ = 0, then | z 1 | + | z 2 | = (1 / | α | ) { ( | - β + αγ | ) + ( | - β - αγ | ) } . ] 10. Show that the necessary and sufficient conditions that both the roots of the equation z 2 + az + b = 0 should be of unit modulus are | a | 5 2 , | b | = 1 , am b = 2 am a. [The amplitudes have not necessarily their principal values.] 11. If x 4 +4 a 1 x 3 +6 a 2 x 2 +4 a 3 x + a 4 = 0 is an equation with real coefficients and has two real and two complex roots, concyclic in the Argand diagram, then a 2 3 + a 2 1 a 4 + a 3 2 - a 2 a 4 - 2 a 1 a 2 a 3 = 0 .
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[III : 49] COMPLEX NUMBERS 124 12. The four roots of a 0 x 4 +4 a 1 x 3 +6 a 2 x 2 +4 a 3 x + a 4 = 0 will be harmonically related if a 0 a 2 3 + a 2 1 a 4 + a 3 2 - a 0 a 2 a 4 - 2 a 1 a 2 a 3 = 0 . [Express Z 23 , 14 Z 31 , 24 Z 12 , 34 , where Z 23 , 14 = ( z 1 - z 2 )( z 3 - z 4 )+( z 1 - z 3 )( z 2 - z 4 ) and z 1 , z 2 , z 3 , z 4 are the roots of the equation, in terms of the coefficients.] 13. Imaginary points and straight lines. Let ax + by + c = 0 be an equation with complex coefficients (which of course may be real in special cases). If we give x any particular real or complex value, we can find the correspond- ing value of y . The aggregate of pairs of real or complex values of x and y which satisfy the equation is called an imaginary straight line ; the pairs of values are called imaginary points , and are said to lie on the line . The values of x and y are called the coordinates of the point ( x, y ). When x and y are real, the point is called a real point : when a , b , c are all real (or can be made all real by division by a common factor), the line is called a real line . The points x = α + βi , y = γ + δi and x = α - βi , y = γ - δi are said to be conjugate ; and so are the lines ( A + A 0 i ) x + ( B + B 0 i ) y + C + C 0 i = 0 , ( A - A 0 i ) x + ( B - B 0 i ) y + C - C 0 i = 0 . Verify the following assertions:—every real line contains infinitely many pairs of conjugate imaginary points; an imaginary line in general contains one and only one real point; an imaginary line cannot contain a pair of conjugate imaginary points:—and find the conditions ( a ) that the line joining two given imaginary points should be real, and ( b ) that the point of intersection of two imaginary lines should be real. 14. Prove the identities ( x + y + z )( x + 3 + 2 3 )( x + 2 3 + 3 ) = x 3 + y 3 + z 3 - 3 xyz, ( x + y + z )( x + 5 + 4 5 )( x + 2 5 + 3 5 )( x + 3 5 + 2 5 )( x + 4 5 + 5 ) = x 5 + y 5 + z 5 - 5 x 3 yz + 5 xy 2 z 2 . 15. Solve the equations x 3 - 3 ax + ( a 3 + 1) = 0 , x 5 - 5 ax 3 + 5 a 2 x + ( a 5 + 1) = 0 . 16. If f ( x ) = a 0 + a 1 x + · · · + a k x k , then { f ( x ) + f ( ωx ) + · · · + f ( ω n - 1 x ) } /n = a 0 + a n x n + a 2 n x 2 n + · · · + a λn x λn ,
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[III : 49] COMPLEX NUMBERS 125 ω being any root of x n = 1 (except x = 1), and λn the greatest multiple of n contained in k . Find a similar formula for a μ + a μ + n x n + a μ +2 n x 2 n + . . . .
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