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The minimum distance of $\text{z}_1 = 3 + 4i$ from $|\text{z} - 1 - i| = 2$ equals
Option: 1
$\sqrt{13}-2$

Option: 2
$\sqrt{13}+2$

Option: 3
$\sqrt{41}+2$

Option: 4
$\sqrt{41}-2$

Answers (1)

best_answer

As we have learnt in

Equation of Circle:

The equation of the circle whose center is at the point $z_0$ and have radius r is given by

$|z-z_0| = r$

If the center is origin then, $z_0=0$ , hence equation reduces to |z| = r

Interior of the circle is represented by $|z-z_0| < r$

The exterior is represented by $|z-z_0| > r$

Here z can be represented as x + iy and $z_0$ is represented by $x_0 + iy_0$

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|z - 1 - i| = 2 is the equation of a circle with centre (1,1) and radius = 2 units.

$z_1 = 3 + 4i$ is a point (3,4), the minimum distance between them will be “distance between point (3,4) and (1,1) minus radius”, so

$\\\mathrm{Minimum \;distance = \sqrt{(3-1)^2 + (4-1)^2}-2} \\\mathrm{=\sqrt{4+9}-2=\sqrt{13}-2}$

so the correct option is (a)

Posted by

Shailly goel

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