Question

Assertion :If $|z+\frac{1}{2}|=1$, where $z,(z\ne 0)$ is a complex number, then the maximum value of $z$ is $\frac{\sqrt{5}+1}{2}$. Reason: On the locus $|z+\frac{1}{z}|=1$, the farthest distance from origin is $\frac{1+\sqrt{5}}{2}$.

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A).
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
• C. (A) is true (R) is false.
• D. (A) is false (R) is true.

Answer 1

The correct option is A Both (A) & (R) are individually true & (R) is correct explanation of (A).

Write $|z|=|z+\frac{1}{z}-\frac{1}{z}|\le |z+\frac{1}{z}|+\frac{1}{|z|}\le 1+\frac{1}{|z|}$
$\Rightarrow |z{|}^2-|z|-1\le 0$
$\Rightarrow (|z|-\frac{1}{2}{)}^2-(\frac{\sqrt{5}}{2})\le 0$
$\Rightarrow (|z|-\frac{1}{2}-\frac{1}{2}+\frac{\sqrt{5}}{2})(|z|-\frac{1}{2}-\frac{\sqrt{5}}{2})\le 0$
$\Rightarrow \frac{1-\sqrt{5}}{2}\le |z|\le \frac{1+\sqrt{5}}{2}$
$\Rightarrow 0<|z|\le \frac{1+\sqrt{5}}{2}(\because |z|>0)$
$\therefore$ Maximum value of |z| is $\frac{1+\sqrt{5}}{2}$
$\Rightarrow$ Assertion is true and Reason is true and Reason is correct explanation of Assertion.