Question 5, Exercise 1.2

Solutions of Question 5 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

Question 5

If z1 and z2 are two any complex numbers then prove that |z1+z2|2−|z1−z2|2=4Re(z1)Re(z2)FIXME

Solution. Suppose z1=x1+iy1 and z2=x2+iy2 Now z1+z2=x1+iy1+x2+iy2=x1+x2+i(y1+y2)|z1+z2|2=(x1+x2)2+(y1+y2)2=x21+x22+2x1x2+y21+y22+2y1y2...(1) z1−z2=x1+iy1−(x2+iy2)=x1−x2+i(y1−y2)|z1−z2|2=(x1−x2)2+(y1−y2)2=x21+x22−2x1x2+y21+y22−2y1y2...(2) Now from (1) and (2) |z1+z2|2−|z1−z2|2=(x21+x22+2x1x2+y21+y22+2y1y2)−(x21+x22−2x1x2+y21+y22−2y1y2)=2x1x2+2y1y2+2x1x2+2y1y2=4x1x2+4y1y2=4Re(z1)Re(z2)+4Im(z1)Im(z2).

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