Question

The measure of the acute angle between the lines given by the equation \( 3x^2 - 4\sqrt{3}xy + 3y^2 = 0 \) is

• A. \( 45^\circ \)
• B. \( 60^\circ \)
• C. \( 70^\circ \)
• D. \( 30^\circ \)

Hint:

For equations of the form \( ax^2 + 2hxy + by^2 = 0 \), always use the standard tangent formula to find the angle between the lines.

Answer 1

The Correct Option is D

Step 1: Identify the general form.
The given equation represents a pair of straight lines through the origin of the form \[ ax^2 + 2hxy + by^2 = 0 \] Comparing, we get \[ a = 3, \quad 2h = -4\sqrt{3}, \quad b = 3 \] Step 2: Use the angle formula.
The angle \( \theta \) between the two lines is given by \[ \tan \theta = \left| \frac{2\sqrt{h^2 - ab}}{a + b} \right| \] Step 3: Substitute the values.
\[ h^2 = 12, \quad ab = 9 \] \[ \tan \theta = \frac{2\sqrt{12 - 9}}{3 + 3} \] Step 4: Simplify.
\[ \tan \theta = \frac{2\sqrt{3}}{6} = \frac{\sqrt{3}}{3} \] Step 5: Find the angle.
\[ \theta = 30^\circ \] Step 6: Conclusion.
The measure of the acute angle between the given lines is \( 30^\circ \).