Question

If $11z^{10}+10i{z}^9+10iz-11=0then$

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

The equation can be rewritten as $z^9=\frac{11-10iz}{11z+10i}$
If z = a + ib then
$\left|z\right|{}^9=\sqrt{\frac{{11}^2+220b+{10}^2(a^2+b^2)}{{11}^2(a^2+b^2)+220b+{10}^2}}$
Let f(a, b) and g(a, b) denote the numerator and denominator of the RHS.
If $\left|z\right|>$ 1 then $a^2+b^2$ $\ge$ 1 so $g(a,b)>$ f(a, b)
Leading to $\left|z\right|{}^9<1$ a contradiction
If $\left|z\right|<$ 1, then $a^2+b^2<1$ so $g(a,b)<f(a,b)$ fielding $\left|z^9\right|>1$ again a contradiction.
Hence $\left|z\right|=1$