Problem 12 - Algebra and Trigonometry

Question

Write the appropriate rotation formulas so that in a rotated system the equation has no \(x^{\prime} y^{\prime}\) -term. $$7 x^{2}-6 \sqrt{3} x y+13 y^{2}-16=0$$

Step-by-Step Solution

Answer

The rotation angle can be found by solving the equation \(\tan(2\theta) = \frac{B}{A - C}\) derived from the expanded equation, where A, B, and C are coefficients from the original equation of conic section.

Step 1: Recognize the equation and rotation of axes

The equation given is of a conic section, which has general form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). To remove the \(x^{\prime} y^{\prime}\) term, a rotation of axes is needed. The rotation formulas are \(x = x^{\prime}\cos(\theta) - y^{\prime}\sin(\theta)\) and \(y = x^{\prime}\sin(\theta) + y^{\prime}\cos(\theta)\).

Step 2: Apply the rotation formulas

Rotating the axes would lead to a new equation \(A(x^{\prime}\cos(\theta) - y^{\prime}\sin(\theta))^2 + B(x^{\prime}\cos(\theta) - y^{\prime}\sin(\theta))(x^{\prime}\sin(\theta) + y^{\prime}\cos(\theta)) + C(x^{\prime}\sin(\theta) + y^{\prime}\cos(\theta))^2 + Dx^{\prime} + Ey^{\prime} + F = 0\). Expand and simplify this equation.

Step 3: Find the rotation angle

Notice that the coefficient in front of \(x^{\prime}y^{\prime}\) term should vanish to satisfy the original requirement. This gives an equation for \(\tan(2\theta)\). Solve this equation and find the rotation angle \(\theta\) to eliminate the \(x^{\prime}y^{\prime}\) term.