Problem 13 - Algebra and Trigonometry

Question

Write the appropriate rotation formulas so that in a rotated system the equation has no \(x^{\prime} y^{\prime}\) -term. $$10 x^{2}+24 x y+17 y^{2}-9=0$$

Step-by-Step Solution

Answer

The rotation formulas for the given quadratic equation are \(x = x' cos(\theta) - y' sin(\theta)\) and \(y = x' sin(\theta) + y' cos(\theta)\), where \(\theta = 1/2 atan(\frac{24}{10-17})\).

Step 1: Understand the Rotation Formulas

The rotation matrix in 2D space is given by the formula: \[ \begin{pmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta) \end{pmatrix} \]\n\nIn a rotated coordinate system where \(x = x' cos(\theta) - y' sin(\theta)\) and \(y = x' sin(\theta) + y' cos(\theta)\), we would replace \(x\) and \(y\) in our given equation with these transformations.

Step 2: Substitute the Rotated Variables into the Equation

Let's substitute: \(x = x' cos(\theta) - y' sin(\theta)\) and \(y = x' sin(\theta) + y' cos(\theta)\) into our equation \(10 x^{2}+24 x y+17 y^{2}-9=0\).

Step 3: Simplify the Equation

Now we need to simplify the equation by expanding and gathering terms. At this stage, we can't simplify it completely because we still don't know the value of \(\theta\). However, we want to set up the equation such that we eliminate the mixed term \(x'y'\). The condition for the \(x'y'\) term to be zero (considering the coefficients) which will give us the value for \(\theta\) is: \((a-d)sin(2\theta) = b cos(2\theta)\), where a, b, d are coefficients of \(x^2\), \(xy\), \(y^2\) in our function respectively.

Step 4: Solve for Theta

Now we can solve for \(\theta\). This will involve rearranging the equation from step 3 to express \(\theta\) in terms of the constants in the original equation. Solving for \(\theta\), we get \(\theta = 1/2 atan(\frac{b}{a-d})\).

Step 5: Substitute Theta Back

The final step involves substituting the value of \(\theta\) obtained from step 4 into the equations from step 2. These will be the final set of rotation formulas for the given quadratic equation.