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Chapter 9: Problem 37

Determine the angle of rotation necessary to transform the equation in \(x\) and \(y\) into an equation in \(X\) and \(Y\) with no \(X Y\) -term. $$3 x^{2}+10 x y+5 y^{2}-1=0$$

Short Answer

Expert verified
Angle of rotation is approximately \(-38.66^\circ.\)

Step by step solution

01

Identify the Coefficients

For the given quadratic equation \(3x^2 + 10xy + 5y^2 - 1 = 0\), identify the coefficients of the terms: \(a = 3\), \(b = 10\), and \(c = 5\). Here, \(b\) is the coefficient of the \(xy\)-term.
02

Set Up the Rotation Angle Formula

To eliminate the \(XY\)-term, use the formula for the rotation angle \(\theta\), given by \(\tan(2\theta) = \frac{b}{a-c}\). Substitute \(b = 10\), \(a = 3\), and \(c = 5\) into this formula.
03

Calculate the Rotation Angle

Compute \(\tan(2\theta)\) with the values from Step 2: \(\tan(2\theta) = \frac{10}{3-5} = \frac{10}{-2} = -5\). Now, calculate \(2\theta\) and consequently \(\theta\) such that \(\tan(2\theta) = -5\).
04

Determine \(\theta\)

To find \(\theta\), solve the equation \(2\theta = \tan^{-1}(-5)\). Using a calculator, find \(\tan^{-1}(-5)\) and then divide the result by 2 to get \(\theta\). This calculation yields \(\theta \approx -38.66^\circ\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coordinate Transformation
A coordinate transformation is a method used to change the coordinates of a geometric object to a different system. This can be thought of as altering the viewpoint from which the object is seen. Mathematically, it involves replacing one set of coordinates with another.

For example, consider a coordinate plane with x- and y-axes, and you wish to change to a new system, X and Y. This new system might be rotated relative to the original. Transforming coordinates involves using equations to relate these two systems.
  • Point Transformation: If a point in the old system is (x, y), its new coordinates could be (X, Y) based on the transformation used.
  • Rotation: A common type of transformation involves rotating the plane by an angle.
    For this, trigonometric functions and the angle of rotation typically come into play.
The goal is often to simplify an equation, such as eliminating the cross-product term (XY term) in quadratic expressions.
Quadratic Equations
Quadratic equations are a fundamental part of mathematics and appear in many forms. They usually are expressions of the form ax虏 + bxy + cy虏 + ... = 0. In this form, they describe a conic section, like circles, ellipses, parabolas, or hyperbolas.

Key points about quadratic equations:
  • Variables: Often, x and y are the common variables in these equations.
  • Terms and Coefficients: Coefficients such as a, b, and c are assigned to terms like x虏, xy, and y虏.
    These coefficients dictate the nature and orientation of the conic section.
  • Objective: Sometimes, the aim is to simplify the equation by removing specific terms, such as the xy-term, which rotational transformations can achieve.
Quadratics are used extensively in algebra, calculus, and many real-world applications, because they often model trajectories, optimization problems, and physical phenomena like gravity.
Rotation Angle Formula
The rotation angle formula is a critical piece of the puzzle when you want to eliminate mixed terms in a quadratic equation. This task involves rotating the coordinate axes by a specific angle, effectively simplifying the equation.

The main step in this process is calculating the angle of rotation. For quadratic equations featuring an xy-term, this can be deduced using the formula: \[ \tan(2\theta) = \frac{b}{a-c} \]
  • Significance: This formula helps in finding the angle \( \theta \), which allows the transformation into a cleaner form of the equation, without the cross-product term.
  • Solving: Once \( \tan(2\theta) \) is known, using inverse trigonometric functions like \( \tan^{-1} \) permits finding \( 2\theta \). Dividing by 2 then yields \( \theta \).
  • Practical Application: With \( \theta \) obtained, a coordinate transformation is applied, aligning the new axes perfectly to remove the xy-term.
The formula and calculations are essential in ensuring that the resultant equation is easier to handle, making problems more manageable.

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Most popular questions from this chapter

Graph the second-degree equation. (Hint: Transform the equation into an equation that contains no \(x y\) -term.) $$21 x^{2}+10 \sqrt{3} x y+31 y^{2}-144=0$$

Identify the conic section with equation \(y^{2}+a x^{2}=x\) for \(a<0, a>0, a=0,\) and \(a=1\)

The ratio of the sum of two numbers to the difference of the two numbers is \(9 .\) The product of the two numbers is 80. Find the numbers.

In calculus, when finding the derivative of equations in two variables, we typically use implicit differentiation. A more direct approach is used when an equation can be solved for one variable in terms of the other variable. In Exercises \(77-80\), solve each equation for \(y\) in terms of \(x\). $$3 x y=-x^{3} y^{2}, y<0$$

Graph the second-degree equation. (Hint: Transform the equation into an equation that contains no \(x y\) -term.) $$71 x^{2}-58 \sqrt{3} x y+13 y^{2}+400=0$$
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