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Magazine Chapter 10: Problem 20
Graph the equation. $$3 x^{2}+10 x y+3 y^{2}+8=0$$
Short Answer
Expert verified
The graph of the equation is a rotated hyperbola.
Step by step solution
01
- Identify the type of conic section
The given equation is in the general form of a conic section: \[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\]. In this case, we have: \[A = 3, B = 10, C = 3, D = 0, E = 0, F = 8\]. Since \[B^2 - 4AC = 10^2 - 4 \cdot 3 \cdot 3 = 100 - 36 = 64\], and 64 > 0, the conic section is a hyperbola.
02
- Simplify the given equation
Rewrite the equation to identify its properties better:\[3x^2 + 10xy + 3y^2 + 8 = 0\]. The equation is in its simplified form.
03
- Rotate the coordinate system
To remove the \(xy\)-term, the system is rotated by an angle \(\theta\) where \(\tan(2\theta) = \frac{B}{A - C}\):\[\tan(2\theta) = \frac{10}{3 - 3} = \infty\]. Thus, \(\theta = \frac{\pi}{4}\).
04
- Substitute the new coordinates
Using the rotation formulas:\[x = x'\cos(\theta) - y'\sin(\theta)\] and \[y = x'\sin(\theta) + y'\cos(\theta)\]. For \(\theta = \frac{\pi}{4}\):\[x = \frac{1}{\sqrt{2}}(x' - y')\] and \[y = \frac{1}{\sqrt{2}}(x' + y')\]. Substitute these into the equation.
05
- Simplify the rotated equation
Substitute back and simplify:\[3 \left( \frac{(x' - y')}{\sqrt{2}} \right)^2 + 10 \left( \frac{(x' - y')}{\sqrt{2}} \right) \left( \frac{(x' + y')}{\sqrt{2}} \right) + 3 \left( \frac{(x' + y')}{\sqrt{2}} \right)^2 + 8 = 0\]which simplifies to a standard hyperbola form.
06
- Final equation and graph
The simplified equation resembles:\[(x')^2 - (y')^2 = -\frac{8}{3}\].Graph this hyperbola by plotting its asymptotes and vertices in the rotated coordinate system and transferring them back to the original coordinates.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Conic Sections
Conic sections are curves formed by the intersection of a plane and a double-napped cone. There are four types: circles, ellipses, parabolas, and hyperbolas. They are represented by a general second-degree polynomial equation: Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 In our problem, the coefficients A, B, and C determine the type of conic section. For instance, with the equation 3x虏 + 10xy + 3y虏 + 8 = 0, we can use B虏 - 4AC to identify the conic: B虏 - 4AC B虏 > 4AC (Hyperbola) Notice that a hyperbola has two branches opening in opposite directions.
Coordinate Rotation
Coordinate rotation is a technique used to simplify conic sections by eliminating the xy-term. This makes recognizing the shape and graphing easier. The rotation angle 胃 is found using: 胃 = 0.5 arctan(B / (A - C)) In our problem with A = 3, B = 10, and C = 3, we get 胃 = 蟺/4. With this angle, the new coordinates are: x = x'cos(胃) - y'sin(胃) y = x'sin(胃) + y'cos(胃) This transformation allows us to simplify the given equation into a more recognizable form.
Simplifying Equations
Simplifying an equation helps to reveal the conic section's properties. After the coordinate rotation, our equation is: 3 *((x' - y')/鈭�2)虏 + 10 * ((x' - y')/鈭�2) * ((x' + y')/鈭�2) + 3 * ((x' + y')/鈭�2)虏 + 8 = 0 Simplify the expression step-by-step: 3(x'虏 - 2x'y' + y'虏)/2 + 10(x'虏 - y'虏)/2 + 3(x'虏 + 2x'y' + y'虏)/2 + 8 = 0 Further simplification reveals: 8(x'虏 - y'虏) = -8 Resulting in: (x')虏 - (y')虏 = -8/3 This is the standard form of a hyperbola equation.
Hyperbola Properties
A hyperbola's properties stem from its definition and equation. For our equation (x')虏 - (y')虏 = -8/3, the key properties include: Asymptotes: Lines that the hyperbola approaches but never intersects, defined by y = 卤x after rotation. Vertices: Points where the hyperbola crosses the x-axis in the rotated system, located at 卤鈭�(8/3). Transferring these vertices and asymptotes back to the original coordinate system gives the complete graph.
