question_answer

For all complex numbers \[{{z}_{1}},{{z}_{2}}\]  satisfying \[|{{z}_{1}}|\,=12\,\] \[\,\text{and }\,|{{z}_{2}}-3-4i|\,=5,\] the minimum value of \[|{{z}_{1}}-{{z}_{2}}|\] is [IIT Screening 2002]

A) 0

B) 2

C) 7

D) 17

Correct Answer: B

Solution :

The two circles are \[{{C}_{1}}(0,\,0),\,{{r}_{1}}=12\], \[{{C}_{2}}(3,\,4),\,{{r}_{2}}=5\] and it passes through origin, the centre of \[{{C}_{1}}\]. \[{{C}_{1}}{{C}_{2}}=5<{{r}_{1}}-{{r}_{2}}=7\] Hence circle \[{{C}_{2}}\]lies inside circle\[{{C}_{1}}\]. Therefore minimum distance between them is \[AB={{C}_{1}}B-{{C}_{1}}A={{r}_{1}}-2{{r}_{2}}=12-10=2.\]