The problem asks to rotate the axes so that the $xy$-term of the conic equation is eliminated, and to find the angle of rotation for that transformation. The given equation is:

$7x^2 + 8xy + 7y^2 - 12 = 0$

Step 1: General form of conic equation

The general second-degree equation for conics is:

$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$

In this case:

• $A = 7$
• $B = 8$
• $C = 7$

Step 2: Formula for angle of rotation

To eliminate the $xy$-term, we need to rotate the axes. The angle $\theta$ of rotation that eliminates the $xy$-term is given by:

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

Substitute the values for $A$, $B$, and $C$:

$\tan(2\theta) = \frac{8}{7 - 7} = \frac{8}{0}$

Since the denominator is zero, this implies:

$\tan(2\theta) \to \infty$

This occurs when $2\theta = 90^\circ$, or:

$\theta = 45^\circ$

Step 3: Final Answer

The smallest positive angle of rotation is $\theta = 45^\circ$.

Would you like more details on how the rotation affects the equation, or should I help you graph the transformed equation?

Here are some additional questions related to this problem:

1. How does rotating the axes affect the shape of a conic section?
2. What is the significance of eliminating the $xy$-term in conic sections?
3. How would you perform a rotation of axes on more complex conic sections?
4. What are some other methods to simplify conic equations?
5. How does the discriminant of a conic section change after a rotation?

Tip: When the $B$ term is non-zero in a conic equation, rotating the axes can simplify the equation and help identify the conic's type.