Problem 10 - Precalculus: Mathematical for Calculus

Question

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

The conic equation after rotation is \(X^2 + Y^2 = 6\).

Step 1: Identify the Original Conic Equation

The conic equation given in terms of \(x\) and \(y\) is \(x^2 + 2y^2 = 16\). This represents an ellipse because the coefficients of \(x^2\) and \(y^2\) are both positive and different.

Step 2: Determine the Rotation Matrix

The rotation angle \(\phi\) is given as \(\sin^{-1}\left(\frac{3}{5}\right)\). Calculate \(\cos \phi = \sqrt{1 - \sin^2 \phi} = \frac{4}{5}\) (since \(\cos^2 \phi + \sin^2 \phi = 1\)). The rotation matrix \(R\) is \[ R = \begin{bmatrix} \cos \phi & -\sin \phi \ \sin \phi & \cos \phi \end{bmatrix} = \begin{bmatrix} \frac{4}{5} & -\frac{3}{5} \ \frac{3}{5} & \frac{4}{5} \end{bmatrix}. \]

Step 3: Express New Coordinates

The new coordinates \((X, Y)\) can be obtained from old coordinates \((x, y)\) using the rotation matrix: \[ \begin{bmatrix} X \ Y \end{bmatrix} = \begin{bmatrix} \frac{4}{5} & -\frac{3}{5} \ \frac{3}{5} & \frac{4}{5} \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix}. \] This results in the transformation equations: \( x = \frac{4}{5}X - \frac{3}{5}Y \) and \( y = \frac{3}{5}X + \frac{4}{5}Y \).

Step 4: Substitute and Simplify

Substitute these expressions for \(x\) and \(y\) into the original conic equation: \(\left(\frac{4}{5}X - \frac{3}{5}Y\right)^2 + 2\left(\frac{3}{5}X + \frac{4}{5}Y\right)^2 = 16\). Expanding and simplifying this expression gives a new equation in \(X\) and \(Y\).

Step 5: Calculate Resulting Equation

Expand the expressions: \(\left(\frac{16}{25}X^2 - \frac{24}{25}XY + \frac{9}{25}Y^2\right) + 2\left(\frac{9}{25}X^2 + \frac{24}{25}XY + \frac{16}{25}Y^2\right) = 16\). Simplifying, \(\frac{34}{25}X^2 + \frac{34}{25}Y^2 = 16\). Multiplying through by \(25\), we get \(34X^2 + 34Y^2 = 400\). Dividing by \(2\), we find \(X^2 + Y^2 = \frac{400}{68}\).

Step 6: Finalize the Simplified Equation

Finally, simplify the equation to \(X^2 + Y^2 = \frac{200}{34}\) or \(X^2 + Y^2 = 6\). This is the equation of the conic in \(X,Y\) coordinates after the rotation.

Key Concepts

coordinate rotation

Coordinate rotation is a mathematical technique used to simplify or change the perspective of a geometric problem. It involves turning the coordinate axes by a certain angle to represent the shape in a new system of coordinates. This can help make equations or problems easier to understand and solve. In our exercise, the original conic equation \(x^2 + 2y^2 = 16\) needed to be rotated through an angle \(\phi\) to transform the ellipse into new coordinates \(X, Y\).- The angle \(\phi\) is the rotation angle, provided here as \(\sin^{-1}\left(\frac{3}{5}\right)\). The values of \(\sin \phi\) and \(\cos \phi\) are essential.- A rotation matrix named \(R\) is used to make this rotation. It is integral because it transforms the original coordinates \((x, y)\) to new coordinates \((X, Y)\).- The transformation helps us see the conic section more clearly from different axes, making it simpler to analyze or further compute the problem.

ellipse

An ellipse is a conic section formed when a plane cuts through a cone at an angle that is not perpendicular to the cone's base. The equation \(x^2 + 2y^2 = 16\) represents an ellipse, a shape characterized by its elongated circle-like structure.- Ellipses have two main axes—it has a major axis (the longest diameter) and a minor axis (the shortest diameter).- Ellipses are defined by their **center**, **foci**, and **intercepts**. In our problem, the center remains at the origin because the equation doesn't include linear terms with \(x\) and \(y\).- The given equation indicates that the ellipse's axes align with the coordinate axes before rotation. When axes are rotated, the new equation \(X^2 + Y^2 = 6\) forms another ellipse aligned differently in the new coordinate system.
The transformation shifts how the ellipse relates spatially, but the geometric properties like area stay consistent.

trigonometric identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the involved variables. These identities are crucial in solving geometric problems involving angles.- In our exercise, using identities such as \(\cos^2 \phi + \sin^2 \phi = 1\) allows us to compute \(\cos \phi\) once \(\sin \phi\) is known.- For the angle \(\phi = \sin^{-1}\left(\frac{3}{5}\right)\), identifying \(\cos \phi\) was essential to find the correct rotation matrix.- The identities simplify calculus and geometry problems because they transform the angles needed for calculations in a manageable form.- Knowledge of identities underlines the problem-solving steps for rotating coordinate systems or computing alternative geometric perspectives.