a+ib≤c+id is meaningful only when:(a)a²+c²=0,(b)b²+c²=0,(c)b²+d²=0,(d)a²+d²=0.

Asked by yeeshuraj3 | 22nd Jul, 2016, 06:39: PM

Expert Answer:

In space complex space number space straight z subscript 1 less than straight z subscript 2 space or space space straight z subscript 1 space greater than space straight z subscript 2 space are space meaningless space but space open vertical bar straight z subscript 1 close vertical bar greater than open vertical bar straight z subscript 2 close vertical bar space space or space open vertical bar straight z subscript 1 close vertical bar less than open vertical bar straight z subscript 2 close vertical bar space space are space meaninggul.

Let space open vertical bar straight a plus ib close vertical bar less or equal than open vertical bar straight c plus id close vertical bar................... left parenthesis straight i right parenthesis
rightwards double arrow open vertical bar straight a plus ib close vertical bar squared less or equal than open vertical bar straight c plus id close vertical bar squared
rightwards double arrow left parenthesis straight a plus ib right parenthesis left parenthesis straight a minus ib right parenthesis less-than or slanted equal to left parenthesis straight c plus id right parenthesis left parenthesis straight c minus id right parenthesis
rightwards double arrow straight a squared minus straight b squared less or equal than straight c squared minus straight d squared
rightwards double arrow straight a squared plus straight d squared less-than or slanted equal to straight c squared plus straight b squared

In space the space above space expression space all space the space terms space are space square space of space straight a space number space hence space all space the space terms space are space greater space than space or space equal space to space zeo.
rightwards double arrow straight a squared greater-than or slanted equal to space 0 space space or space space straight d squared space greater-than or slanted equal to space 0 space space space or space straight c squared greater-than or slanted equal to space 0 space space space or space space straight b squared space space greater-than or slanted equal to space 0

Now comma space option space left parenthesis straight a right parenthesis space and space left parenthesis straight c right parenthesis space are space of space no space use.
space
For space option space left parenthesis straight b right parenthesis
If space space space straight b squared plus straight c squared equals 0 space then space we space are space left space with space no space chose space but space straight a squared plus straight d squared space equals space 0
Which space means
straight a space equals space straight b space equals space straight c space equals space straight d space equals space 0
which space means space open vertical bar straight a plus ib close vertical bar space equals open vertical bar straight c plus id close vertical bar space which space contradicts space space eq space left parenthesis 1 right parenthesis

For space option space left parenthesis straight d right parenthesis
If space straight a squared plus straight d squared space equals space 0 space space then space we space can space have space either space straight b squared plus straight c squared equals 0 space or space straight b squared plus straight c squared greater than 0
Which space means
which space means space either space open vertical bar straight a plus ib close vertical bar space equals open vertical bar straight c plus id close vertical bar space space or space space open vertical bar straight a plus ib close vertical bar space less than open vertical bar straight c plus id close vertical bar
Hence space option space left parenthesis straight d right parenthesis space space is space space correct.

Answered by Vijaykumar Wani | 25th Jul, 2016, 11:53: AM