Problem 17 - Algebra and Trigonometry

Question

Write the appropriate rotation formulas so that in a rotated system the equation has no \(x^{\prime} y^{\prime}\) -term. $$34 x^{2}-24 x y+41 y^{2}-25=0$$

Step-by-Step Solution

Answer

The rotation formulas are \(x' = xcos(\Theta) - ysin(\Theta)\) and \(y' = xsin(\Theta) + ycos(\Theta)\) with calculated \(\Theta\). After the substitution of \(x'\) and \(y'\) into the equation and simplification, the equation will be transformed into without the \(x'y'\) term.

Step 1: Understand and Define Rotation Formulas

To make the equation have no \(x'y'\)-term, the standard rotation formulas can be applied. They are:\[\begin{align*}x' = xcos(\Theta) - ysin(\Theta)\y' = xsin(\Theta) + ycos(\Theta)\end{align*}\]

Step 2: Determine the Angle of Rotation

The angle of rotation (\(\Theta\)) can be found by using the formula:\[tg(2\Theta) = \frac{2b}{a-c}\]where \(a\), \(b\) and \(c\) are coefficients before \(x^2\), \(xy\) and \(y^2\) respectively. Thus, \[\Theta = \frac{1}{2}arctg(\frac{-2b}{a-c}) \]Substitute \(b=-12\), \(a=34\) and \(c=41\) into the formula to find the angle of rotation.

Step 3: Substitute Angle of Rotation into Rotation Formulas

Using the calculated angle of rotation, substitute the values of \(\Theta\) back into the rotation formulas from step 1 so that \(x'\) and \(y'\) are each expressed in terms of \(x\) and \(y\).

Step 4: Substitute \(x'\) and \(y'\) into Conic Section Equation

Replace \(x'\) and \(y'\) with the equivalent rotation formulas from step 3 into the given conic section equation and simplify the equation to transform it into the form without term \(x'y'\).