If z is a complex number such that |z| geq 1, | vedclass

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(D) Let $z_0 = -\frac{1}{2}(3+4i) = -\frac{3}{2} - 2i$. We want to find the minimum value of $|z - z_0|$,where $|z| \geq 1$. Geometrically,this repres

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$\frac{3}{2}$

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(D) Let $z_0 = -\frac{1}{2}(3+4i) = -\frac{3}{2} - 2i$. We want to find the minimum value of $|z - z_0|$,where $|z| \geq 1$. Geometrically,this represents the minimum distance from a point $z$ on or outside the unit circle $|z|=1$ to the fixed point $z_0 = -\frac{3}{2} - 2i$. The distance of the point $z_0$ from the origin is $|z_0| = \sqrt{(-\frac{3}{2})^2 + (-2)^2} = \sqrt{\frac{9}{4} + 4} = \sqrt{\frac{25}{4}} = \frac{5}{2}$. Since the point $z_0$ lies outside the unit circle $(|z_0| = 2.5 > 1)$,the minimum distance from the circle $|z|=1$ to the point $z_0$ is given by $|z_0| - r$,where $r=1$ is the radius of the unit circle. Minimum value $= |z_0| - 1 = \frac{5}{2} - 1 = \frac{3}{2}$.

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

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