Question:
If the angle between the pair of lines $x^2 - 2cxy - 7y^2 = 0$ is $\frac{\pi}{3}$, then the value of $c^2$ is:
If the angle between the pair of lines $x^2 - 2cxy - 7y^2 = 0$ is $\frac{\pi}{3}$, then the value of $c^2$ is:
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Simplify the fraction inside the angle formula first before squaring to prevent arithmetic mistakes: $\sqrt{3} = \frac{\sqrt{c^2+7}}{3} \implies c^2+7 = 27 \implies c^2 = 20$.
Updated On: May 31, 2026
- $20$
- $10$
- $5$
- $15$
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The Correct Option is A
Solution and Explanation
Step 1: Concept
The angle $\theta$ between the pair of straight lines represented by the equation $ax^2 + 2hxy + by^2 = 0$ is given by: \[ \tan\theta = \frac{2\sqrt{h^2 - ab}}{|a + b|} \]
Step 2: Meaning
Comparing the given equation $x^2 - 2cxy - 7y^2 = 0$ with the general equation, we get $a = 1$, $b = -7$, $2h = -2c \implies h = -c$, and the angle $\theta = \frac{\pi}{3}$.
Step 3: Analysis
Substitute the parameters into the formula: \[ \tan\left(\frac{\pi}{3}\right) = \frac{2\sqrt{(-c)^2 - (1)(-7)}}{|1 - 7|} \] \[ \sqrt{3} = \frac{2\sqrt{c^2 + 7}}{|-6|} \] \[ \sqrt{3} = \frac{2\sqrt{c^2 + 7}}{6} = \frac{\sqrt{c^2 + 7}}{3} \] Cross-multiplying and squaring both sides: \[ 3\sqrt{3} = \sqrt{c^2 + 7} \implies 27 = c^2 + 7 \implies c^2 = 20 \]
Step 4: Conclusion
The value of $c^2$ is $20$. Final Answer: (A)
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