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Mathematics Question on Complex Numbers and Quadratic Equations

If z1=2−i,z2=1+i,find ∣z1+z2+1z1−z2+1∣.If \,z_1=2-i,z_2=1+i, find \,|\frac{z_1+z_2+1}{z1-z2+1}|.

Answer

z1=2−i,z2=1+iz_1=2-i,z_2=1+i

∣z1+z2+1z1−z2+1=∣(2−i)+(1+i)+1(2−i)+(1+i)+1|\frac{z_1+z_2+1}{z1-z2+1}=|\frac{(2-i)+(1+i)+1}{(2-i)+(1+i)+1}

∣42−2i=∣42(1−i)|\frac{4}{2-2i}=|\frac{4}{2(1-i)}

=∣21−i×1+i1+i∣=∣2(1+i)12−i2∣=|\frac{2}{1-i}×\frac{1+i}{1+i}|=|\frac{2(1+i)}{1^2-i^2}|

=2(1+i)1+i)∣=\frac{2(1+i)}{1+i)}| [i2=−1][i^2=-1]

∣2(1+i)2∣|\frac{2(1+i)}{2}|

∣1+i∣=12+12=2|1+i|=\sqrt1^2+1^2=\sqrt2

Thus,the value of ∣z1+z2+1z1−z2+1] is 2|\frac{z_1+z_2+1}{z_1-z_2+1}]\,is\,\sqrt2.