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:

• A. $20$
• B. $10$
• C. $5$
• D. $15$

Hint:

Simplify the fraction inside the angle formula first before squaring to prevent arithmetic mistakes: $\sqrt{3} = \frac{\sqrt{c^2+7}}{3} \implies c^2+7 = 27 \implies c^2 = 20$.

Answer 1

The Correct Option is A

Step 1: Angle between a pair of lines.
For $ax^2 + 2hxy + by^2 = 0$, the angle satisfies $\tan\theta = \frac{2\sqrt{h^2 - ab}}{|a + b|}$.

Step 2: Match the coefficients.
From $x^2 - 2cxy - 7y^2 = 0$: $a = 1$, $b = -7$, and $2h = -2c$ so $h = -c$.

Step 3: Put in the angle.
With $\theta = \frac{\pi}{3}$, $\tan\theta = \sqrt3$: \[ \sqrt3 = \frac{2\sqrt{c^2 + 7}}{|1 - 7|} \]

Step 4: Simplify the bottom.
$|1 - 7| = 6$, so \[ \sqrt3 = \frac{2\sqrt{c^2 + 7}}{6} = \frac{\sqrt{c^2 + 7}}{3} \]

Step 5: Clear and square.
\[ 3\sqrt3 = \sqrt{c^2 + 7} \implies 27 = c^2 + 7 \]

Step 6: Solve.
\[ c^2 = 20 \] \[ \boxed{ c^2 = 20 } \]