We are tasked with eliminating the $xy$-term of the given conic equation:

$10x^2 - 4\sqrt{3}xy + 6y^2 - 36 = 0$

Step 1: General formula for rotation of axes

The angle of rotation $\theta$ to eliminate the $xy$-term in the equation of a conic is given by:

$\tan(2\theta) = \frac{B}{A - C}$

where:

• $A$ is the coefficient of $x^2$,
• $B$ is the coefficient of $xy$,
• $C$ is the coefficient of $y^2$.

For the given equation:

• $A = 10$,
• $B = -4\sqrt{3}$,
• $C = 6$.

Step 2: Plug into the formula

Substitute $A = 10$, $B = -4\sqrt{3}$, and $C = 6$ into $\tan(2\theta) = \frac{B}{A - C}$:

$\tan(2\theta) = \frac{-4\sqrt{3}}{10 - 6} = \frac{-4\sqrt{3}}{4} = -\sqrt{3}.$

Step 3: Solve for $2\theta$

From $\tan(2\theta) = -\sqrt{3}$, recall that $\tan(\phi) = -\sqrt{3}$ corresponds to angles of $120^\circ$ or $300^\circ$ (in standard position, one in the second quadrant and one in the fourth quadrant).

Thus: $2\theta = 120^\circ \quad \text{or} \quad 2\theta = 300^\circ.$

Divide by 2 to find $\theta$: $\theta = 60^\circ \quad \text{or} \quad \theta = 150^\circ.$

Since we need the smallest positive angle, the angle of rotation is:

$\theta = 60^\circ.$

Final Answer:

The angle of rotation to eliminate the $xy$-term is: $\boxed{60^\circ}.$

Would you like help analyzing the rotated conic, graphing it, or further explaining the rotation process? Let me know!

Related Questions:

1. What are the steps to rewrite the conic equation after rotation?
2. How do we determine the type of conic (ellipse, hyperbola, parabola) from the coefficients?
3. What is the general matrix method for eliminating the $xy$-term in conics?
4. How do we graph conic sections after rotation?
5. Can the rotation method apply to degenerate conics?

Tip: Remember that conic sections with rotation often require converting coordinates to their new axes for graphing!