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Chapter 10: Problem 13

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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 \(10 x^{2}+24 x y+17 y^{2}-9=0\).
03

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.
04

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 \(\theta = 1/2 atan(\frac{b}{a-d})\).
05

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.

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Most popular questions from this chapter

If you are given the standard form of the equation of a parabola with vertex at the origin, explain how to determine if the parabola opens to the right, left, upward, or downward.

Describe one similarity and one difference between the graphs of \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\) and \(\frac{(x-1)^{2}}{25}+\frac{(y-1)^{2}}{16}=1\)

a. Identify the conic section that each polarequation represents. b. Describe the location of a directrix from the focus located at the pole. $$ r=\frac{8}{2-2 \sin \theta} $$

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. $$ r=\frac{8}{1+\cos \theta} $$

a. Identify the conic section that each polarequation represents. b. Describe the location of a directrix from the focus located at the pole. $$ r=\frac{12}{2+4 \cos \theta} $$
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