Question

Let $s, t, r$ be non-zero distinct positive real numbers. If the complex number $z=x+iy$ satisfies $sz+t\overline{z}+r=0$, then $z$ lies on:

• A. imaginary axis
• B. real axis
• C. $y=x$
• D. $y=2x$
• E. $x+y=0$

Hint:

A complex number with an imaginary part of zero always lies on the real axis.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We are given an equation involving a complex number \( z \), its conjugate \( \bar{z} \), and some real constants. We need to find the locus (the geometric path) of the point \( (x, y) \) corresponding to \( z \) in the complex plane.
Step 2: Key Formula or Approach:
The standard approach is to substitute \( z = x + iy \) and \( \bar{z} = x - iy \) into the given equation and then separate the real and imaginary parts. A complex number is equal to zero if and only if both its real and imaginary parts are zero.
Step 3: Detailed Explanation:
The given equation is \( sz + t\bar{z} + r = 0 \).
Substitute \( z = x + iy \) and \( \bar{z} = x - iy \):
\[ s(x + iy) + t(x - iy) + r = 0 \] Distribute s and t:
\[ sx + siy + tx - tiy + r = 0 \] Group the real terms and the imaginary terms:
\[ (sx + tx + r) + i(sy - ty) = 0 \] This can be written as \( (s+t)x + r + i(s-t)y = 0 \).
For this complex number to be equal to zero, both its real part and its imaginary part must be zero.
Real Part: \( (s+t)x + r = 0 \)
Imaginary Part: \( (s-t)y = 0 \)
From the imaginary part, we have \( (s-t)y = 0 \).
The problem states that s and t are distinct positive real numbers, which means \( s \neq t \), and therefore \( s - t \neq 0 \).
Since the product \( (s-t)y \) is zero and \( s-t \) is non-zero, it must be that \( y = 0 \).
Step 4: Final Answer:
The condition \( y = 0 \) means that the imaginary part of the complex number \( z = x + iy \) is always zero. Complex numbers with a zero imaginary part lie on the real axis in the complex plane.