Question:
The equation $|z+i|-|z-i|=k$ represents a hyperbola, if
The equation $|z+i|-|z-i|=k$ represents a hyperbola, if
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The equation $|z+i|-|z-i|=k$ represents a hyperbola, if
Updated On: Apr 15, 2026
- $-2<k<2$
- $k>2$
- $0<k<2$
- None of these
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The Correct Option is A
Solution and Explanation
Step 1: Concept
The equation $|z-z_1| - |z-z_2| = k$ represents a hyperbola if $k$ is less than the distance between the fixed points $z_1$ and $z_2$.
Step 2: Analysis
Here, the fixed points are $z_1 = -i$ and $z_2 = i$. The distance between these two points on the imaginary axis is $|i - (-i)| = |2i| = 2$.
Step 3: Evaluation
For the locus to be a hyperbola, the constant $k$ must satisfy the condition $|k|<\text{distance between foci}$.
Step 4: Conclusion
Thus, $|k|<2$, which means $-2<k<2$.
Final Answer: (a)
The equation $|z-z_1| - |z-z_2| = k$ represents a hyperbola if $k$ is less than the distance between the fixed points $z_1$ and $z_2$.
Step 2: Analysis
Here, the fixed points are $z_1 = -i$ and $z_2 = i$. The distance between these two points on the imaginary axis is $|i - (-i)| = |2i| = 2$.
Step 3: Evaluation
For the locus to be a hyperbola, the constant $k$ must satisfy the condition $|k|<\text{distance between foci}$.
Step 4: Conclusion
Thus, $|k|<2$, which means $-2<k<2$.
Final Answer: (a)
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