Question:
If $\frac{z-1}{2z+1}$ is an imaginary number and if it represents a circle then its radius is
If $\frac{z-1}{2z+1}$ is an imaginary number and if it represents a circle then its radius is
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For purely imaginary $(z-a)/(z-b)$, the points $z$ lie on a circle with diameter joining $a$ and $b$.
Updated On: May 14, 2026
- $\frac{9}{16}$ units
- $\frac{3}{4}$ units
- $\frac{1}{4}$ units
- $\frac{1}{2}$ units
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The Correct Option is B
Solution and Explanation
Step 1: Concept
A complex number is purely imaginary if its real part is zero.
Step 2: Meaning
Let $z = x + iy$. Then $\text{Re}\left(\frac{x+iy-1}{2(x+iy)+1}\right) = 0$.
Step 3: Analysis
Rationalize the fraction: $\frac{(x-1)+iy}{(2x+1)+2iy} \cdot \frac{(2x+1)-2iy}{(2x+1)-2iy}$. Real part $= (x-1)(2x+1) + 2y^2 = 0$. $2x^2 + x - 2x - 1 + 2y^2 = 0 \implies 2x^2 - x + 2y^2 = 1 \implies x^2 - x/2 + y^2 = 1/2$.
Step 4: Conclusion
Complete the square: $(x - 1/4)^2 + y^2 = 1/2 + 1/16 = 9/16$. Radius $= \sqrt{9/16} = 3/4$. Final Answer: (B)
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