question_answer

DIRECTION (Qs. 80): Each of these questions contains two statements: Statement-1 (Assertion) and Statement-2 (Reason). Choose the correct the answer (Only one option is correct) from the following-

Statement-1: If \[|{{z}_{1}}|=30,|{{z}_{2}}-(12+5i)|=6,\] 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, \[{{z}_{1}}\]are \[{{z}_{2}}\]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, Statement-2 is true, and Statement-2 is a correct explanation for Statement-1

C)  Statement-1 is true, Statement-2 is true; Statement-2 is not a 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.