question_answer

Statement 1: If \[|{{z}_{1}}|=30,|{{z}_{2}}-(12+5i)=6,\] then maximum value of\[|{{z}_{1}}-{{z}_{2}}|\] is 49.

Statement 2: If \[{{z}_{1}},{{z}_{2}}\] are two complex numbers, then \[|{{z}_{1}}-{{z}_{2}}|\le |{{z}_{1}}|+|{{z}_{2}}|\]and equality holds when origin, \[{{z}_{1}}\]and \[{{z}_{2}}\] are collinear and \[{{z}_{1}},{{z}_{2}}\] are on the opposite side of the origin.

A)  Statement-1 is false, Statement-2 is true.

B)  Statement-1 is true, statemerit-2 is true and Statement-2 is correct explanation for statement-1

C)  Statement-1 is true, Statement-2 is true and Statement-2 is NOT correct explanation for statement-1

D)  Statement-1 is true, Statement-2 is false.

Correct Answer: C

Solution :

 \[{{C}_{1}}{{C}_{2}}=13\]\[{{r}_{1}}=30,{{r}_{2}}=6\]\[{{C}_{1}}{{C}_{2}}<{{r}_{1}}-{{r}_{2}}\] \[\therefore \]The circle\[|{{z}_{2}}-(12+5i)|=6\]lies within the circle\[|{{z}_{1}}|=30\] \[\therefore \]\[\max |{{z}_{1}}-{{z}_{2}}|=30+13+6=49\] \[\therefore \]Statement-1 is true. Statement-2\[|{{z}_{1}}-{{z}_{2}}|\le |{{z}_{1}}|+|{{z}_{2}}|\]is always true. Equality sign holds if \[{{z}_{1}},{{z}_{2}}\]origin are collinear and \[{{z}_{1}}\]and\[{{z}_{2}}\]lies on opposite sides of the origin. \[\therefore \]Statement-2 is true.