Problem 12 - Algebra and Trigonometry

Question

\(9-14\) . Determine the equation of the given conic in \(X Y\) -coordinates when the coordinate axes are rotated through the indicated angle. $$ x^{2}+2 y^{2}=16, \quad \phi=\sin ^{-1} \frac{3}{5} $$

Step-by-Step Solution

Answer

After rotation, the conic equation becomes a new quadratic in terms of \(X\) and \(Y\).

Step 1: Identify the starting equation and angle

We are given the equation of a conic: \( x^{2}+2y^{2}=16 \). The angle of rotation \( \phi \) is given as \( \sin^{-1}\left(\frac{3}{5}\right) \). This means \( \sin \phi = \frac{3}{5} \). We can find \( \cos \phi \) using the Pythagorean identity, where \( \cos^2 \phi + \sin^2 \phi = 1 \).

Step 2: Calculate \( \cos \phi \)

Since \( \sin \phi = \frac{3}{5} \), we have:\[\cos \phi = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5}\]Thus, \( \cos \phi = \frac{4}{5} \).

Step 3: Write the rotation matrix

The rotation matrix for the rotation by angle \( \phi \) is given by:\[\begin{pmatrix}\cos \phi & \sin \phi \-\sin \phi & \cos \phi\end{pmatrix} = \begin{pmatrix}\frac{4}{5} & \frac{3}{5} \-\frac{3}{5} & \frac{4}{5}\end{pmatrix}\]We will use this matrix to perform the coordinate transformation.

Step 4: Apply the rotation to the conic equation

The new coordinates \((X, Y)\) after rotation are related to the original coordinates \((x, y)\) by:\[X = x \cos \phi + y \sin \phi \Y = -x \sin \phi + y \cos \phi\]Substitute these into the original conic equation \( x^{2} + 2y^{2} = 16 \).

Step 5: Perform substitution and simplify

After substitutions, you will analyze the coefficients that result from expanding:1. Express \( x \) and \( y \) in terms of \( X \) and \( Y \) using the inverse of the rotation matrix.2. Substitute these expressions into \( x^{2} + 2y^{2} = 16 \).3. Simplify the equation to get an expression solely in terms of \( X \) and \( Y \).

Key Concepts

Understanding the Rotation of Axes

When you need to rotate the axes in a coordinate system, you're shifting the perspective from which you view a shape, like a conic section, on a plane. This allows you to express equations that might have cross terms more cleanly, without these additional terms that can complicate analysis. The angle of rotation is denoted by \(\phi\), which in our exercise is \(\sin^{-1}\left(\frac{3}{5}\right)\). This means that you're tilting the axes by an angle at which the sine is \(\frac{3}{5}\).
The calculation of \(\cos \phi\) uses the identity \(\cos^2 \phi + \sin^2 \phi = 1\), helping us relate sine and cosine effectively. With \(\cos \phi = \frac{4}{5}\), we have all we need for a rotation. Rotating the axes often simplifies the equation of the conic by aligning it closer to our axes, thus minimizing off-axis terms that make solving and interpreting the equation harder.

Coordinate Transformation Technique

Coordinate transformation involves replacing one set of coordinates with another, to facilitate easier analysis or simplification of equations. In our case, the transformation is a result of the rotation of axes.
The relationship between the new coordinates \((X, Y)\) and the old ones \((x, y)\) is given by:

• \(X = x \cos \phi + y \sin \phi\)
• \(Y = -x \sin \phi + y \cos \phi\)

This transformation is applied using the rotation matrix. The matrix helps in expressing every point's new position by considering both rotation and potential translation in the plane. It's a powerful tool for rewriting each term of the conic section in terms of the new coordinates, making subsequent algebra easier.

Algebraic Equation Simplification

Simplifying algebraic equations after transformation is a crucial step for clearer and more useful results. Post transformation, substitute \(x\) and \(y\) in terms of \(X\) and \(Y\) into the given equation of the conic: \(x^2 + 2y^2 = 16\).
This involves expanding the equations, combining like terms, and reorganizing the expression. The goal is to express everything in terms of the new coordinates \((X, Y)\), eliminating any remaining dependence on \(x\) and \(y\).
Through careful substitution and algebraic manipulation, you achieve a cleaner form of the equation that describes the conic section relative to its new, rotated position. This simplified equation often makes it easier to determine properties of the conic, such as its center, axes lengths, and orientation.