| Directions (Q. Nos. 86) The geometrical meaning of \[|{{z}_{1}}-{{z}_{2}}|,\] where \[{{z}_{1}}\] and \[{{z}_{2}}\] are points in Argand plane is the distance between the points \[{{z}_{1}}\] and \[{{z}_{2}}\] based on this information, a class of problems about least value can be solved. The property that the sum of two sides of a triangle is greater than the third side is also very useful in solving these problems. |
A) \[1+\sqrt{7}\]
B) \[\sqrt{13}\]
C) \[2+\sqrt{13}\]
D) \[\sqrt{7}+\sqrt{13}\]
Correct Answer: B
Solution :
\[|z+i|\,+|z+3i|+|\,-z+2|+|-z-7i|\] \[\ge \,|z+i\,+z+3i-z+2-z-7i|\] \[=\,|2-3i|\,=\,\sqrt{13}\] \[(\because \,\,|{{z}_{1}}|\,+\,|{{z}_{2}}|\,+|{{z}_{3}}|\,+|{{z}_{4}}|\,\,\ge \,\,{{z}_{1}}+{{z}_{2}}+{{z}_{3}}+{{z}_{4}}|)\]
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