# Complex number- finding the least possible value

• Aug 21st 2006, 02:55 AM
kingkaisai2
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.
• Aug 21st 2006, 05:00 AM
CaptainBlack
Quote:

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
• Aug 21st 2006, 08:31 AM
CaptainBlack
Quote:

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
• Aug 21st 2006, 10:57 AM
Soroban
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"?

Quote:

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.