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Magazine Chapter 10: Problem 10
Rotate the coordinate axes to remove the \(xy-term.\) Then identify the type of conic and sketch its graph. $$5 x^{2}-6 x y+5 y^{2}-8 \sqrt{2} x+8 \sqrt{2} y=8$$
Short Answer
Expert verified
The equation represents a circle after rotation.
Step by step solution
01
Identify the Coefficients of the Quadratic Part
Look at the equation given: \[ 5x^2 - 6xy + 5y^2 - 8\sqrt{2}x + 8\sqrt{2}y = 8 \]. Here, the coefficients are \( A = 5 \), \( B = -6 \), and \( C = 5 \). These will be used in calculating the angle of rotation.
02
Calculate the Angle of Rotation
The angle \( \theta \) for rotation to remove the \( xy \)-term is given by the formula: \[ \tan(2\theta) = \frac{B}{A - C} \].Substitute the values: \[ \tan(2\theta) = \frac{-6}{5 - 5} \]. This simplifies to an undefined value, suggesting \( \theta = 45^\circ \), as the tangent function is vertical.
03
Apply the Rotation Formulas
The coordinate transformation due to rotation is:\[ x = X \cos(\theta) - Y \sin(\theta) \] \[ y = X \sin(\theta) + Y \cos(\theta) \].Here, \( \theta = 45^\circ \) gives \( \cos(\theta) = \sin(\theta) = \frac{\sqrt{2}}{2} \).Transforming gives:\[ x = \frac{1}{\sqrt{2}}(X - Y) \] \[ y = \frac{1}{\sqrt{2}}(X + Y) \].
04
Substitute and Simplify the Transformed Equation
Replace the values of \( x \) and \( y \) in the given equation using the above transformations:\[ 5\left(\frac{1}{\sqrt{2}}(X - Y)\right)^2 - 6 \left(\frac{1}{\sqrt{2}}(X - Y)\right)\left(\frac{1}{\sqrt{2}}(X + Y)\right) + 5\left(\frac{1}{\sqrt{2}}(X + Y)\right)^2 \. - 8\sqrt{2}\left(\frac{1}{\sqrt{2}}X - \frac{1}{\sqrt{2}}Y \right) + 8\sqrt{2}\left(\frac{1}{\sqrt{2}}X + \frac{1}{\sqrt{2}}Y \right) = 8\].Simplify terms for \(X^2, Y^2\) and constants. The middle term, \(-6xy\), vanishes. This simplifies to \[ 10X^2 + 0XY + 10Y^2 - 16X + 16Y = 8 \].
05
Identify the Type of Conic
The equation can be rewritten as:\[ 10(X^2 + Y^2) - 16X + 16Y = 8 \], which is a circle equation in disguise (complete the square to confirm). Thus, the conic is a circle.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Rotation of Axes
When dealing with conic sections, sometimes the presence of an \(xy\)-term can complicate the equation. Removing it makes equations easier to handle. One useful method is rotation of axes. This involves rotating the coordinate system around the origin by some angle \(\theta\).
The goal is to find this angle so that the new equation no longer has the mixed term \(xy\). To determine \(\theta\), the formula used is \(\tan(2\theta) = \frac{B}{A - C}\). If this formula gives an undefined value, which happens when the denominator is zero, then \(\theta\) is typically \(45^\circ\) or \(135^\circ\). These angles often simplify complex situations, such as in our given problem where the original term \( -6xy \) becomes zero.
It鈥檚 crucial to substitute this rotation angle into rotation formulas:
The goal is to find this angle so that the new equation no longer has the mixed term \(xy\). To determine \(\theta\), the formula used is \(\tan(2\theta) = \frac{B}{A - C}\). If this formula gives an undefined value, which happens when the denominator is zero, then \(\theta\) is typically \(45^\circ\) or \(135^\circ\). These angles often simplify complex situations, such as in our given problem where the original term \( -6xy \) becomes zero.
It鈥檚 crucial to substitute this rotation angle into rotation formulas:
- \( x = X\cos(\theta) - Y\sin(\theta) \)
- \( y = X\sin(\theta) + Y\cos(\theta) \)
Conic Identification
After removing the \(xy\)-term using a rotation, the transformed equation must be analyzed to identify the type of conic section it represents. Conic sections鈥攃ircles, ellipses, parabolas, and hyperbolas鈥攁re determined by the presence of specific coefficients.
Following a rotation, if the transformed equation of the form \(AX^2 + C Y^2 + DX + EY = F\) has equal coefficients \(A\) and \(C\) and no \(XY\) term, this often indicates a circle.
In our exercise, the presence of equal \(X^2\) and \(Y^2\) coefficients in the equation \(10X^2 + 10Y^2 - 16X + 16Y = 8\) precisely points to a circle. This is because both \(A\) and \(C\) are equal, satisfying the circle's properties.
Recognizing these patterns helps in sketching accurate graphs of these conic sections.
Following a rotation, if the transformed equation of the form \(AX^2 + C Y^2 + DX + EY = F\) has equal coefficients \(A\) and \(C\) and no \(XY\) term, this often indicates a circle.
In our exercise, the presence of equal \(X^2\) and \(Y^2\) coefficients in the equation \(10X^2 + 10Y^2 - 16X + 16Y = 8\) precisely points to a circle. This is because both \(A\) and \(C\) are equal, satisfying the circle's properties.
Recognizing these patterns helps in sketching accurate graphs of these conic sections.
Equation Transformation
Transforming equations aligns them with standard forms, making it simpler to understand and graph mathematical expressions. In our case, the transformation of the original conic equation into \(10(X^2 + Y^2) - 16X + 16Y = 8\) without the \(XY\) term is a good example.
The rotation removes complexity by eliminating the \(xy\) term, thus reducing the equation to one of standard form like \((X-a)^2 + (Y-b)^2 = r^2 \) for circles.
This facilitated transformation through rotation of axes simplifies analysis and drawing.
Practice this method frequently; not only does it reveal the conic's nature, but it also makes sketching and further manipulations straightforward.
Working through these transformations reinforces understanding of geometry and algebra's interconnectedness.
The rotation removes complexity by eliminating the \(xy\) term, thus reducing the equation to one of standard form like \((X-a)^2 + (Y-b)^2 = r^2 \) for circles.
This facilitated transformation through rotation of axes simplifies analysis and drawing.
Practice this method frequently; not only does it reveal the conic's nature, but it also makes sketching and further manipulations straightforward.
Working through these transformations reinforces understanding of geometry and algebra's interconnectedness.
