# Thread: Complex number- finding the least possible value

1. ## Complex number- finding the least possible value

find the least possible value of [z+2-sqrt3i] <2. Describe in geometrical terms.
sqrt= square root - it is the square root of 3.

2. Originally Posted by kingkaisai2
find the least possible value of [z+2-sqrt3i] <2. Describe in geometrical terms.
sqrt= square root - it is the square root of 3.
Reformulate this question so that it actualy asks something.

RonL

3. Originally Posted by kingkaisai2
find the least possible value of [z+2-sqrt3i] <2. Describe in geometrical terms.
sqrt= square root - it is the square root of 3.
Part of this maybe should be: Describe |z+2-sqrt3i| <2 in geometric terms.

As before this defines the set of all points in the complex plane interior of a circle of radius 2 centred at -2+i.sqrt(3).

RonL

4. Hello, kingkaisai2!

The problem is badly worded ... it makes no sense.
I'll take a guess at what you meant . . .

Also, in all your problems, unless the inequality is $\displaystyle \leq$
. . there is no solution.
Are you sure it's "less than"?

Find the least possible value of $\displaystyle |z|$ such that $\displaystyle |z + 2 - i\sqrt{3}| \:\leq \:2$
Describe in geometrical terms.

We have: .$\displaystyle |z - (-2 + i\sqrt{3})| \:\leq \:2$

This is the set of points within 2 units of $\displaystyle -2 + i\sqrt{3}.$

This is the set of points in or on the circle
. . with center $\displaystyle C(-2,\sqrt{3})$ and radius $\displaystyle r = 2.$
Code:
                          |
* * *       |
*           *   |
*               * |
*                 *|
_       |
*      (-2,√3)      *
*        C*         *
*           \       *
\     |
*              \  *|
*               o |
-----*-----------*---+----
* * *       |O
|

The point $\displaystyle z$ with the least magnitude
. . is the intersection of the circle and the line $\displaystyle CO.$

The line has the equation: .$\displaystyle y \:= \:-\frac{\sqrt{3}}{2}x$

The circle has the equation: .$\displaystyle (x + 2)^2 + (y - \sqrt{3})^2\:=\:4$

Find their intersections and take the rightmost point.

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### find the least possible value of modulus z

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