Question
For all complex numbers $${z_1},{z_2}$$ satisfying $$\left| {{z_1}} \right| = 12\,\,{\text{and }}\left| {{z_2} - 3 - 4i} \right| = 5,$$ the minimum value of $$\left| {{z_1} - {z_2}} \right|$$ is
A.
0
B.
2
C.
7
D.
17
Answer :
2
Solution :
$$\left| {{z_1}} \right| = 12$$
$$ \Rightarrow \,{z_{1\,}}\,$$ lies on a circle with center $$\left( {0,0} \right)$$ and radius 12 unites, and $$\left| {{z_2} - 3 - 4i} \right| = 5$$
$$ \Rightarrow \,\,{z_2}$$ lies on a circle with center $$\left( {3,4} \right)$$ and radius 5 units.
From fig. it is clear that $$\left| {{z_1} - {z_2}} \right|$$ i.e., distance between $${{z_1}}$$ and $${{z_2}}$$ will be min when they lie at $$A$$ and $$B$$ resp. i.e., $$O, C, B, A$$ are collinear as shown. Then $${{z_1} - {z_2}}$$ $$= AB = OA$$ $$ - OB = 12 - 2(5) = 2.$$ As above is the min, value, we must have $$\left| {{z_1} - {z_2}} \right| \geqslant 2.$$
$$\left| {{z_1}} \right| = 12$$
$$ \Rightarrow \,{z_{1\,}}\,$$ lies on a circle with center $$\left( {0,0} \right)$$ and radius 12 unites, and $$\left| {{z_2} - 3 - 4i} \right| = 5$$
$$ \Rightarrow \,\,{z_2}$$ lies on a circle with center $$\left( {3,4} \right)$$ and radius 5 units.
From fig. it is clear that $$\left| {{z_1} - {z_2}} \right|$$ i.e., distance between $${{z_1}}$$ and $${{z_2}}$$ will be min when they lie at $$A$$ and $$B$$ resp. i.e., $$O, C, B, A$$ are collinear as shown. Then $${{z_1} - {z_2}}$$ $$= AB = OA$$ $$ - OB = 12 - 2(5) = 2.$$ As above is the min, value, we must have $$\left| {{z_1} - {z_2}} \right| \geqslant 2.$$