Question

solve them for 50 points​

Answer 1

answer of 2nd

A+ib=c+i/c-i

=(c+i)(c+i)/(c-i)(c+i)

=(c²+2ci+i²)/(c²-i²)

=(c²+2ci-1)/{c²-(-1)}

={(c²-1)+2ci}/(c²+1)

∴, a+ib=(c²-1)/(c²+1)+i(2c)/(c²+1)

Equating the real and imaginary part from both sides we get,

a=c²-1/c²+1 and b=2c/c²+1

∴, a²+b²

={(c²-1)/(c²+1)}²+{2c/(c²+1)}²

={(c²-1)²+4c²}/(c²+1)²

=(c⁴-2c²+1+4c²)/(c²+1)²

=(c⁴+2c²+1)/(c²+1)²

=(c²+1)²/(c²+1)²

=1 (Proved)

Now, b/a

={2c/(c²+1)}/{(c²-1)/(c²+1)}

=2c/(c²+1)×(c²+1)/(c²-1)

=2c/(c²-1) (Proved)

Answer 2

$\sf{\large {\underline {COMPLEX\:NUMBERS}}}$ :

REFER THE ATTACHMENT.