Question:
If the equation $ax^{2}+2hxy+by^{2}+2gx+2fy+c=0$ represents a pair of parallel lines, then $g^{2}h^{2}=$
If the equation $ax^{2}+2hxy+by^{2}+2gx+2fy+c=0$ represents a pair of parallel lines, then $g^{2}h^{2}=$
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Remember the parallel line coefficient ratio shortcut: $af = hg$, which leads directly to $a^2f^2 = g^2h^2$ upon squaring.
Updated On: Jun 3, 2026
- $a^{2}b^{2}$
- $a^{2}$
- $f^{2}$
- $a^{2}f^{2}$
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The Correct Option is D
Solution and Explanation
Step 1: Concept
For a general second-degree equation to represent a pair of parallel lines, the second-degree terms must form a perfect square, which requires $h^2 = ab$. Additionally, the condition for the lines to be parallel implies $\frac{a}{h} = \frac{h}{b} = \frac{g}{f}$.
Step 2: Meaning
From the relation $\frac{a}{h} = \frac{g}{f}$, we can cross-multiply to find a direct relationship between the coefficients: $af = hg$.
Step 3: Analysis
Squaring both sides of the condition $hg = af$ gives: $(hg)^2 = (af)^2 \implies h^2g^2 = a^2f^2$.
Step 4: Conclusion
Rearranging the terms, we get $g^2h^2 = a^2f^2$, which perfectly matches option (D).
Final Answer: (D)
For a general second-degree equation to represent a pair of parallel lines, the second-degree terms must form a perfect square, which requires $h^2 = ab$. Additionally, the condition for the lines to be parallel implies $\frac{a}{h} = \frac{h}{b} = \frac{g}{f}$.
Step 2: Meaning
From the relation $\frac{a}{h} = \frac{g}{f}$, we can cross-multiply to find a direct relationship between the coefficients: $af = hg$.
Step 3: Analysis
Squaring both sides of the condition $hg = af$ gives: $(hg)^2 = (af)^2 \implies h^2g^2 = a^2f^2$.
Step 4: Conclusion
Rearranging the terms, we get $g^2h^2 = a^2f^2$, which perfectly matches option (D).
Final Answer: (D)
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