Question:
The equation $\frac{x^2}{7-k}+\frac{y^2}{5-k}=1$ represents a hyperbola if}
The equation $\frac{x^2}{7-k}+\frac{y^2}{5-k}=1$ represents a hyperbola if}
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Hyperbola = Opposite signs in denominators; Ellipse = Same signs in denominators.
Updated On: Jul 4, 2026
- $5 < k < 7$
- $k > 5$
- $k < 5$ or $k > 7$
- $k \neq 5, k \neq 7$
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The Correct Option is A
Solution and Explanation
Step 1: Concept A general equation $x^2/A + y^2/B = 1$ represents a hyperbola if the signs of $A$ and $B$ are opposite.
Step 2: Meaning The product of denominators must be negative: $(7-k)(5-k) < 0$.
Step 3: Analysis This inequality holds when $k$ lies between the roots of the expression.
Step 4: Conclusion The roots are $k=5$ and $k=7$. Thus, the condition is $5 < k < 7$. Final Answer: (A)
Step 2: Meaning The product of denominators must be negative: $(7-k)(5-k) < 0$.
Step 3: Analysis This inequality holds when $k$ lies between the roots of the expression.
Step 4: Conclusion The roots are $k=5$ and $k=7$. Thus, the condition is $5 < k < 7$. Final Answer: (A)
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