question_answer

If z lies on the curve arg \[\left( z+i \right)=\frac{\pi }{4},\], then the minimum value of \[|z+4-3i|+|z-4+3i|\] is [Note: \[{{i}^{2}}=-1\]]

A)  5        

B)                                     10

C)  15           

D)                                     20

Correct Answer: B

Solution :

As, z lies on the curve \[\arg (z+i)\,=\frac{\pi }{4},\] which is a ray originating from (-i) and lying right side of imaginary axis making an angle \[\frac{\pi }{4}\] with the real axis anticlockwise sense. \[\therefore \] The value of \[|z-(-4+3i)\,|\,\,+\,\,|\,\,z-(4-3i)\,|\] will minimum when \[z,-4+3i,\text{ }4-3i\]are collinear. \[\therefore \] Minimum value = distance between \[(-4+3i)\] and and \[\left( 4-3i \right)\]             \[=\sqrt{{{(-4-4)}^{2}}\,+{{(3+3)}^{2}}}\,=\sqrt{64\,+36}\,=\sqrt{100}=10\]