Given the condition that the modulus of the complex fraction
|‌| is equal to 1 , we can infer that this fraction lies on the unit circle in the complex plane. This implies that the magnitude of the numerator is equal to the magnitude of the denominator.
So, we have:
|z1+z2|=|z1−z2|Let us write
z1 and
z2 in their general form as complex numbers:
z1=x1+iy1 and
z2=x2+iy2, where
x1,x2,y1, and
y2 are real numbers.
Thus, the equation becomes:
|(x1+x2)+i(y1+y2)|=|(x1−x2)+i(y1−y2)|The modulus of a complex number
a+bi is given by
√a2+b2. Applying this to our equation, we get:
√(x1+x2)2+(y1+y2)2=√(x1−x2)2+(y1−y2)2Squaring both sides, we obtain:
(x1+x2)2+(y1+y2)2=(x1−x2)2+(y1−y2)2Expanding both sides gives:
x12+2x1x2+x22+y12+2y1y2+y22=x12−2x1x2+x22+y12−2y1y2+y22By canceling out the common terms from both sides, we get:
2x1x2+2y1y2=−2x1x2−2y1y2 This further simplifies to:
4x1x2+4y1y2=0Dividing both sides by 4 , we obtain:
x1x2+y1y2=0The expression for the real part of a complex fraction
‌ is given by:
Re(‌)=‌Given that
x1x2+y1y2=0, we substitute this into the expression above:
Re(‌)=‌=0So, we have:
Re(‌)+1=0+1=1