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, Statement-2 is true; 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 \[{{z}_{1}}\,\,\text{and}\]\[{{z}_{2}}\] lies on opposite sides of the origin. \[\therefore \]Statement-2 is true.You need to login to perform this action.
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