Question:
The set of all real values of \( c \) for which the equation
\[
zz' + (4 - 3i)z + (4+3i)z + c = 0
\]
represents a circle is:
The set of all real values of \( c \) for which the equation
\[
zz' + (4 - 3i)z + (4+3i)z + c = 0
\]
represents a circle is:
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A complex number equation represents a circle if it follows the form \( zz' + Az' + A^z + c = 0 \) with \( c \leq |A|^2 \).
Updated On: May 20, 2025
- \( [25, \infty) \)
- \( [-5, 5] \)
- \( (-\infty, -5] \cup [5, \infty) \)
- \( (-\infty, 25] \)
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The Correct Option is D
Solution and Explanation
Step 1: Identify the Circle Condition
The general form of a circle in complex numbers is:
\[
zz' + Az' + A^z + c = 0
\]
where \( A = 4 - 3i \), so:
\[
|A|^2 = (4-3i)(4+3i) = 16 + 9 = 25
\]
Step 2: Condition for a Circle
For the equation to represent a circle,
\[
c \leq |A|^2
\]
\[
c \leq 25
\]
Thus, the correct answer is \( (-\infty, 25] \).
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