Question

Let $z_1$ and $z_2$ be complex numbers satisfying $|z_1|=|z_2|=2$ and $|z_1+z_2|=3$. Then $\left|\frac{1}{z_1}+\frac{1}{z_2}\right|=$

• A. $\frac{3}{2}$
• B. 2
• C. $\frac{3}{4}$
• D. $\frac{1}{2}$
• E. 4

Hint:

Algebra Tip: Whenever you see reciprocals of complex numbers ($\frac{1}{z}$) alongside known moduli ($|z|$), immediately think to use the identity $\frac{1}{z} = \frac{\bar{z}}{|z|^2}$ to clear the denominators!

Answer 1

The Correct Option is C

Concept:
A fundamental property of complex numbers relates a complex number $z$, its complex conjugate $\bar{z}$, and its modulus $|z|$: $z \cdot \bar{z} = |z|^2$. This property is extremely useful for expressing the reciprocal of a complex number: $\frac{1}{z} = \frac{\bar{z}}{|z|^2}$. Also, the modulus of a conjugate is equal to the modulus of the original number: $|\bar{z}| = |z|$.

Step 1: Apply the modulus property to the given complex numbers.
We are given that $|z_1| = 2$ and $|z_2| = 2$. Squaring both sides gives us $|z_1|^2 = 4$ and $|z_2|^2 = 4$. Using the property $z\bar{z} = |z|^2$, we have $z_1\bar{z}_1 = 4$ and $z_2\bar{z}_2 = 4$.

Step 2: Rewrite the reciprocals in terms of their conjugates.
From the equations in Step 1, we can isolate the reciprocals $\frac{1}{z_1}$ and $\frac{1}{z_2}$: $$\frac{1}{z_1} = \frac{\bar{z}_1}{4}$$ $$\frac{1}{z_2} = \frac{\bar{z}_2}{4}$$

Step 3: Substitute these into the target expression.
We need to find the value of $\left|\frac{1}{z_1} + \frac{1}{z_2}\right|$. Substitute our newly found expressions into the modulus: $$\left|\frac{1}{z_1} + \frac{1}{z_2}\right| = \left|\frac{\bar{z}_1}{4} + \frac{\bar{z}_2}{4}\right|$$

Step 4: Factor out the constant from the modulus.
Combine the fractions over the common denominator and pull the constant $\frac{1}{4}$ outside of the absolute value: $$\left|\frac{\bar{z}_1 + \bar{z}_2}{4}\right| = \frac{1}{4} \left|\bar{z}_1 + \bar{z}_2\right|$$

Step 5: Utilize conjugate addition rules and calculate the final answer.
The sum of two conjugates is equal to the conjugate of their sum: $\bar{z}_1 + \bar{z}_2 = \overline{z_1 + z_2}$. Therefore: $$= \frac{1}{4} \left|\overline{z_1 + z_2}\right|$$ Since the modulus of a conjugate is the same as the modulus of the original complex number ($|\bar{z}| = |z|$): $$= \frac{1}{4} |z_1 + z_2|$$ We are given $|z_1 + z_2| = 3$. Substituting this value in yields the final answer: $$= \frac{1}{4} (3) = \frac{3}{4}$$ Hence the correct answer is (C) $\frac{3{4}$}.