Let the complex number Z satisfy the equation |Z + 4i|=3 , then find the greatest and least value of |Z + 3|

lz+4il=3,X^2 +(y+4)^2 =3,this is a circle,z+3=(x+3,y),x^2=3-(y+4)^2, we have to find maximum and minimum values of

(x+3^2 +y^2), use X^2 =3-(y+4)^2 and you will get a single variable equation

For greatest value, use triangle inequality |z1+z2| <=|z1|+|z2|

|z+3| = |(z+4i)+(3-4i)| <=|z+4i| + |3-4i|

  = 3 + 5 = 8

For least value, use triangle inequality, |z1+z2| >=||z1|-|z2||

So, |z+3| = |(z+4i)+(3-4i)| >=||z+4i|-|3-4i||

    = |3-5| = |-2| = 2

So, the greatest value is 8 and least value is 2.

Alternate method of finding the same, is to draw the locus of point z satisfying the condition |z+4i| = 3. This shall be a circle with centre at (0-4i) and radius 3. To find the |Z+3|, plot the point -3+0i in the plane. The distance between the point z in the circle and the point -3+0i gives the value |z+3|. The greatest value shall be 8 and least value shall be 2. (Calculated from geometry)