If z is a complex number such that |z| 1, then the minimum value of |z+(1)/(2)(3+4i)| is :

Question

If $z$ is a complex number such that $|z|\leqslant 1$, then the minimum value of $|z+\frac{1}{2}(3+4i)|$ is :

Accepted Answer

To find the minimum value of $|z+\frac{1}{2}(3+4i)|$, where $z$ is a complex number with $|z|\leqslant 1$, we can think of this geometrically as the distance from any point inside or on the boundary of the unit circle in the complex plane to the fixed point $\frac{1}{2}(3+4i)$.

To make this more clear, first write the expression as follows:
$|z+\frac{1}{2}(3+4i)|=|z+\frac{3}{2}+\frac{4}{2}i|=|z+1.5+2i|$

This means we are looking for the distance between any point on the complex plane represented by $z$ (where $z$ has a magnitude of up to 1) and the point $1.5+2i$.

The triangle inequality gives us the following relationship for any two complex numbers ${z}_{1}$ and ${z}_{2}$:
$|{z}_{1}+{z}_{2}|\geqslant ||{z}_{1}|-|{z}_{2}||$

Applying this to our specific case (${z}_{1}=z$ and ${z}_{2}=-1.5-2i$):
$|z+1.5+2i|\geqslant ||z|-|-1.5-2i||$

Since $z$ is within the unit circle, $|z|\leqslant 1$ and $|-1.5-2i|=\sqrt{{1.5}^{2}+{2}^{2}}=\sqrt{2.25+4}=\sqrt{6.25}=2.5$.

So now we have:
$|z+1.5+2i|\geqslant |1-2.5|$

Which simplifies to:
$|z+1.5+2i|\geqslant 1.5$

Therefore, the minimum value of $|z+\frac{1}{2}(3+4i)|$ given that $|z|\leqslant 1$ is $\frac{3}{2}$.

So the correct answer is:

Option C: $\frac{3}{2}$

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