Problem 88 - A Complete Resource Book in Mathematics for JEE Main

Question

If for the complex numbers \(z_{1}\) and \(z_{2},\left|z_{1}+z_{2}\right|=\left|z_{1}-z_{2}\right|\), then \(\operatorname{amp} z_{1} \sim a m p z_{2}=\) (A) \(\pi\) (B) \(\frac{\pi}{2}\) (C) \(\frac{\pi}{4}\) (D) None of these

Step-by-Step Solution

Answer

The amplitude of \(z_1/z_2\) is \(\frac{\pi}{2}\) (Option B).

Step 1: Understand the Problem

We need to find the amplitude of the ratio of two complex numbers \(z_1\) and \(z_2\) given the condition that the sum and difference of their magnitudes are equal: \( \left|z_1 + z_2\right| = \left|z_1 - z_2\right| \).

Step 2: Use the Given Condition

Given \( \left|z_1 + z_2\right| = \left|z_1 - z_2\right| \), remember that \( |a+b| = |a-b| \) implies that vectors \(a\) and \(b\) are perpendicular. Hence \( z_1 \) and \( z_2 \) are perpendicular.

Step 3: Express Perpendicular Condition Using Angles

If \(z_1\) and \(z_2\) are perpendicular, the angle between them is \(\frac{\pi}{2}\). Thus, \(\operatorname{arg}(z_1) - \operatorname{arg}(z_2) = \frac{\pi}{2}\).

Step 4: Conclusion

Since the condition we derived corresponds to the angle between \(z_1\) and \(z_2\) being \(\frac{\pi}{2}\), the amplitude of the quotient \( z_1/z_2 \) is \(\frac{\pi}{2}\).

Key Concepts

Amplitude of Complex Numbers

The amplitude of a complex number, often referred to as the argument, is a measure of the direction or angle of the number in the complex plane. It helps us understand the placement of the number concerning the positive real axis. Given a complex number in polar form, such as \( z = r(\cos(\theta) + i\sin(\theta)) \), the amplitude is denoted by \( \theta \). The amplitude tells us how much we need to "rotate" the number around the origin.

• When given as an angle, the amplitude can take values ranging from \(-\pi\) to \(\pi\) radians.
• It essentially provides the phase angle of the complex number.
• Understanding amplitude is crucial as it helps us compare and utilize complex numbers effectively, especially in problems involving trigonometric relationships.

Thus, in our given problem, finding the amplitude is vital to establishing the angle relation between the two complex numbers \(z_1\) and \(z_2\).

Magnitude Equality

The concept of magnitude equality is frequently encountered in complex number exercises. When dealing with complex numbers \( z_1 \) and \( z_2 \), if \( \left|z_1 + z_2\right| = \left|z_1 - z_2\right| \), it implies a geometric relationship between these numbers.

• Magnitude refers to the absolute value or the length of the vector represented by a complex number.
• A key takeaway here is that when this magnitude condition holds true, it signifies that the vectors corresponding to the complex numbers form an isosceles triangle with respect to their addition and subtraction.
• This geometric condition indicates specific angular relationships which are pivotal in analyzing complex number problems.

Understanding this helps us determine configurations like perpendicularity, which are otherwise not immediately obvious.

Perpendicular Vectors

Perpendicular vectors in the context of complex numbers offer a fascinating insight into their orientation. For complex numbers, if we interpret them as vectors, being perpendicular means the angle between the two is \( \frac{\pi}{2} \) (90 degrees).

• Two vectors are perpendicular if their dot product equals zero.
• However, with complex numbers, this perpendicularity reflects more straightforwardly through magnitude comparisons, as seen in our problem.
• When \( \left|z_1 + z_2\right| = \left|z_1 - z_2\right| \), it naturally implies the vectors (complex numbers) are perpendicular.
• This subtle relationship lets us infer an angle of \( \frac{\pi}{2} \) between them, leading to a direct conclusion about the amplitude of their ratio.

Thus, recognizing the perpendicular nature of vectors (or complex numbers) is critical to solving many complex plane problems and deducing properties such as amplitudes.