Problem 5 - College Algebra

Question

For the equation \(A x^{2}+B x y+C y^{2}+D x+E y+F=0,\) the value of \(\theta\) that satisfies \(\cot (2 \theta)=\frac{A-C}{B}\) gives us what information?

Step-by-Step Solution

Answer

\(\theta\) aligns the conic's axes with coordinate axes.

Step 1: Understand the equation

The equation given is a general equation of a conic section in terms of the variables \(x\) and \(y\). The coefficients \(A, B, C, D, E,\) and \(F\) determine the type and orientation of the conic.

Step 2: Recognize the angle transformation formula

The value of \(\theta\) such that \(\cot(2\theta) = \frac{A-C}{B}\) comes from the formula to find the angle needed to eliminate the \(xy\)-term in the conic equation. This transforms the conic section to an aligned coordinate system, where the conic aligns with the major and minor axes.

Step 3: Identify the role of \(\theta\)

The angle \(\theta\) indicates the rotation needed to align the coordinate axes with the principal axes of the conic. When \(\theta\) is applied, the \(xy\)-term becomes zero, simplifying the conic into a standard form, like an ellipse, parabola, or hyperbola.

Key Concepts

General Equation of a Conic

The general equation of a conic section is expressed as \(A x^2 + B xy + C y^2 + D x + E y + F = 0\). This equation can represent different types of conic sections such as ellipses, parabolas, and hyperbolas depending on the values of the coefficients \(A, B,\) and \(C\). - If \(B^2 - 4AC < 0\), the conic is an ellipse. - If \(B^2 - 4AC = 0\), it is a parabola. - If \(B^2 - 4AC > 0\), the conic is a hyperbola.
The terms \(D, E,\) and \(F\) in the equation are responsible for translating the conic in the plane but do not affect its shape. Understanding this equation is fundamental, as it determines both the type and orientation of a conic section. The presence of an \(xy\)-term (\(B eq 0\)) indicates that the conic is rotated relative to the coordinate axes, leading us to further transformation considerations.

Angle of Rotation

The concept of angle rotation is crucial for simplifying conic equations that include the \(xy\)-term. By evaluating \(\cot(2\theta) = \frac{A-C}{B}\), we can determine the appropriate rotation angle \(\theta\) needed to eliminate the \(xy\)-term. This transforms the conic equation into a form that is aligned with the principal axes of the coordinate system. - This angle rotation makes calculations easier by simplifying the conic into its standard form. - It highlights how rotation affects the orientation without changing the type of conic.
Once \(\theta\) is found, the coordinate axes can be rotated accordingly. This process of finding \(\theta\) aligns the major and minor axes with the new rotated coordinate axes, making visualization and mathematical manipulation more straightforward. Using this method allows deeper insight into the geometric nature of the conic.

Coordinate Transformation

Coordinate transformation plays a central role in manipulating conic sections. When you rotate the axes by an angle \(\theta\), you essentially change the coordinates to make the equation simpler. Rotating the equation of the conic to remove the \(xy\)-term involves a transformation of the form: - \(x' = x \cos\theta + y \sin\theta\) - \(y' = -x \sin\theta + y \cos\theta\)
Through this change of variables, the conic can be rewritten without the \(xy\)-term, transforming it into a more manageable form. This makes visualizing and analyzing the conic much simpler as the new x' and y' axes align with the conic's principal directions.

• Coordinate transformation aids in understanding the intrinsic properties without direct dependence on the coordinate axes.
• It simplifies the conic to a form that highlights its geometry, making it easier to solve related geometric problems.

Understanding this transformation not only aids in solving conic equations but also clarifies the geometric relationship of the conic relative to its axes.