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: Concept
Substitute $z = x + iy$ and $\overline{z} = x - iy$ into the equation.

Step 2: Analysis
$s(x + iy) + t(x - iy) + r = 0$ $(sx + tx + r) + i(sy - ty) = 0$

Step 3: Analysis
Equating the imaginary part to zero: $(s - t)y = 0$. Since $s$ and $t$ are distinct ($s \neq t$), we must have $y = 0$.

Step 4: Conclusion
Since $y = 0$, the complex number $z$ lies on the real axis. Final Answer: (B)