Question:
The locus of the points represented by \( |z + 3| - |z - 3| = 6 \), where \( z \) is a complex number, is ....
The locus of the points represented by \( |z + 3| - |z - 3| = 6 \), where \( z \) is a complex number, is ....
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For equations of the form \( |z - f_1| - |z - f_2| = d \), the locus is a hyperbola or parabola, depending on the value of \( d \).
Updated On: Jun 30, 2026
- Circle with radius 1 unit
- Straight line with slope 1
- Parabola with focus (1, 0)
- X-axis
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the given equation.
The given equation is:
\[ |z + 3| - |z - 3| = 6 \]
This represents the locus of a point \( z \) in the complex plane, where the difference in distances from two fixed points (here \( -3 \) and \( 3 \)) is constant.
Step 2: Interpreting the geometric figure.
The equation \( |z + 3| - |z - 3| = 6 \) describes a hyperbola, as it is of the form \( |z - f_1| - |z - f_2| = d \), where \( f_1 \) and \( f_2 \) are the foci of the hyperbola and \( d \) is the constant difference of distances.
Step 3: Analyzing the geometry.
The foci of the hyperbola are located at \( (-3, 0) \) and \( (3, 0) \), and the constant difference is 6, so it is a parabola rather than a hyperbola.
Step 4: Conclusion.
Thus, the locus is a parabola with the focus at \( (1, 0) \), which corresponds to option (C).
Final Answer:
The correct answer is: \[ \boxed{\text{Parabola with focus (1, 0)}} \]
The given equation is:
\[ |z + 3| - |z - 3| = 6 \]
This represents the locus of a point \( z \) in the complex plane, where the difference in distances from two fixed points (here \( -3 \) and \( 3 \)) is constant.
Step 2: Interpreting the geometric figure.
The equation \( |z + 3| - |z - 3| = 6 \) describes a hyperbola, as it is of the form \( |z - f_1| - |z - f_2| = d \), where \( f_1 \) and \( f_2 \) are the foci of the hyperbola and \( d \) is the constant difference of distances.
Step 3: Analyzing the geometry.
The foci of the hyperbola are located at \( (-3, 0) \) and \( (3, 0) \), and the constant difference is 6, so it is a parabola rather than a hyperbola.
Step 4: Conclusion.
Thus, the locus is a parabola with the focus at \( (1, 0) \), which corresponds to option (C).
Final Answer:
The correct answer is: \[ \boxed{\text{Parabola with focus (1, 0)}} \]
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