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Chapter 7: Problem 43

Use the Principal Axes Theorem to perform a rotation of axes to eliminate the \(x y\) -term in the quadratic equation. Identify the resulting rotated conic and give its equation in the new coordinate system. $$13 x^{2}-8 x y+7 y^{2}-45=0$$

Short Answer

Expert verified
The rotated conic is an ellipse, with the equation in the new coordinate system being approximately \(3.91x'^2 + 6.92y'^2 =45.\) given that \(x'\) and \(y'\) are coordinates in the rotated system and the approximate value of \(\theta\) is \(-22.39\) degrees, if solutions are rounded to two decimal places.

Step by step solution

01

Find tan(2胃)

To obtain the angle of rotation \(\theta\), the formula tan(2\(\theta\)) = \(\frac{2 h}{a - c}\) is used, where \(h, a,\) and \(c\) are the coefficients from the quadratic equation \(ax^2 + 2hxy + cy^2 = 0\). Here \(a = 13, h = -4\), and \(c = 7\). Substituting these values gives tan(2\(\theta\)) = \(\frac{2 * -4}{13 - 7} = -4\).
02

Calculate the rotation angle 胃

The rotation angle \(\theta\) is half the angle whose tangent is calculated in step 1. It can be calculated using the formula: \(\theta = \frac{1}{2} arctan(\frac{2h}{a - c})\). Therefore, \(\theta = \frac{1}{2} arctan(-4)\). For simplicity, use the approximate value of \(\theta\), which equals -22.390 degrees.
03

Rotate the Axes

Perform the rotation of axes using the transformation formula \(x = rcos(\theta) - ysin(\theta)\), \(y = xsin(\theta) + ycos(\theta)\). Replace \(x\) and \(y\) in the original equation with the transformation equations and simplify. Verify that the \(xy\) term is eliminated in the resulting equation.
04

Identify the Rotated Conic

Once the \(xy\) term is eliminated, the remaining terms represent the equation of a rotated conic. The resulting equation would be in the form of \(ax'^2 + cy'^2 = p\), that implies the conic is an ellipse where \(a\) and \(c\) represent the squares of the lengths of semi-major and semi-minor axes respectively, and \(p\) is a constant.

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