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Chapter 10: Problem 17

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

Short Answer

Expert verified
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 by step solution

01

Understand and Define Rotation Formulas

To make the equation have no \(x'y'\)-term, the standard rotation formulas can be applied. They are:\[\begin{align*}x' = xcos(\Theta) - ysin(\Theta)\y' = xsin(\Theta) + ycos(\Theta)\end{align*}\]
02

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.
03

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 \(y\).
04

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 \(x'y'\).

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Most popular questions from this chapter

Describe a viewing rectangle, or window, such as [-30, 30, 3] by [-8, 4, 1], that shows a complete graph of each polar equation and minimizes unused portions of the screen. $$ r=\frac{8}{1-\cos \theta} $$

Describe a viewing rectangle, or window, such as [-30, 30, 3] by [-8, 4, 1], that shows a complete graph of each polar equation and minimizes unused portions of the screen. $$ r=\frac{16}{5-3 \cos \theta} $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. You told me that an ellipse centered at the origin has vertices at \((-5,0)\) and \((5,0),\) so 1 was able to graph the ellipse.

Exercises 105鈥�107 will help you prepare for the material covered in the next section. Simplify and write the equation in standard form in terms of \(x^{\prime}\) and \(y^{\prime}\) $$ \left[\frac{\sqrt{2}}{2}\left(x^{\prime}-y^{\prime}\right)\right]\left[\frac{\sqrt{2}}{2}\left(x^{\prime}+y^{\prime}\right)\right]=1 $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Although the algebra of rotations can get ugly, the main is that rotation through an appropriate angle will transform a general second-degree equation into an equation in \(x^{\prime}\) and \(y^{\prime}\)
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