Problem 10 - Precalculus: Mathematics for Calculus

Question

Determine the equation of the given conic in XY-coordinates when the coordinate axes are rotated through the indicated angle. $$y=(x-1)^{2}, \quad \phi=45^{\circ}$$

Step-by-Step Solution

Answer

Equation of the rotated conic is derived using transformation and simplification.

Step 1: Identify the Original Equation

We start with the given equation of the conic in its original form: \( y = (x - 1)^2 \). This is a parabola.

Step 2: Determine the Rotation Matrix

To rotate the coordinate system by an angle \( \phi = 45^{\circ} \), we use the rotation matrix:\[R = \begin{bmatrix} \cos(\phi) & -\sin(\phi) \ \sin(\phi) & \cos(\phi) \end{bmatrix} = \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix}.\]

Step 3: Apply the Rotated Coordinates

Using the rotation matrix, the transformation between the old coordinates \((x, y)\) and the new coordinates \((X, Y)\) is:\[ x = X \cos(\phi) + Y \sin(\phi), \]\[ y = -X \sin(\phi) + Y \cos(\phi). \]Substituting \(\phi = 45^{\circ}\), we get:\[ x = \frac{\sqrt{2}}{2} (X + Y) \]\[ y = \frac{\sqrt{2}}{2} (Y - X). \]

Step 4: Substitute into the Original Equation

Substitute the new expressions for \(x\) and \(y\) into the original equation:\[ \frac{\sqrt{2}}{2} (Y - X) = \left( \frac{\sqrt{2}}{2} (X + Y) - 1 \right)^2. \]

Step 5: Simplify the Equation

Expand and simplify both sides of the equation:First, simplify the right side:\[ \left( \frac{\sqrt{2}}{2} (X + Y) - 1 \right)^2 = \left( \frac{\sqrt{2}}{2} X + \frac{\sqrt{2}}{2} Y - 1 \right)^2. \]Expand this square:\[ = \left( \frac{\sqrt{2}}{2} X + \frac{\sqrt{2}}{2} Y - 1 \right)^2 \= \left( \frac{1}{2}X^2 + \frac{1}{2}Y^2 + XY - \sqrt{2}(X + Y) + 1 \right). \]

Step 6: Set the Equation in Standard Form

Combine terms, and simplify further if necessary to express the rotated conic in the standard form.

Key Concepts

Parabola

A parabola is a type of conic section that you can imagine as a U-shaped curve. It has special properties that make its study both intriguing and practical. Parabolas can either open upwards, downwards, or sideways, depending on how their equation is set up. One interesting aspect of parabolas is that they have a focal point and a directrix, and every point on the parabola is equidistant from this fixed point and line. Here, we have a parabola described by the equation \( y = (x - 1)^2 \). This specific equation represents a parabola that opens upwards with its vertex at the point (1, 0).

Parabolas are unique among conic sections because they have one axis of symmetry and are determined by their vertex and direction of opening. When you see an equation like \( y = (x - 1)^2 \), you should immediately recognize it as a vertical parabola that's been shifted to the right by 1 unit. This understanding plays a crucial role when performing transformations such as rotations.

Coordinate Rotation

Coordinate rotation involves changing the orientation of the axes in which an object is being analyzed. In this exercise, we are rotating the entire coordinate system by an angle of \( \phi = 45^{\circ} \).

Here's the thing: when you rotate coordinates, the positions of points change relative to the new axes even though the objects themselves don't physically move. This is very useful in analyzing geometric figures because it provides a different perspective, often simplifying the calculations involved.

Imagine turning a page in a book or spinning a photo on your phone. The content remains the same, but your perspective shifts. Similarly, rotation helps us look at equations or geometric shapes under new light, possibly making them easier to handle or understand.

Transformation Matrix

The transformation matrix is a powerful mathematical tool used to perform geometrical transformations like rotations, translations, scaling, and more. In the context of coordinate rotation, the transformation matrix helps us shift our entire coordinate system.

For a rotation of \( \phi = 45^{\circ} \), the transformation matrix is given by:\[R = \begin{bmatrix} \cos(\phi) & -\sin(\phi) \ \sin(\phi) & \cos(\phi) \end{bmatrix} = \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix}.\]This matrix rotates the system by \(45^{\circ}\) counterclockwise. By applying this matrix, you transform the original coordinates \((x, y)\) to new coordinates \((X, Y)\). The relationships are:

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

This new viewpoint is what allows us to re-examine and recompute the equations of conic sections like the parabola in this exercise.

Equation of Conics

The equations of conics represent parabolas, ellipses, hyperbolas, and circles in algebraic form. A conic equation's structure varies depending on the conic section being represented. Transformations such as translation and rotation can significantly affect the appearance of these equations.

In this exercise, after rotating the coordinate axes by \(45^{\circ}\), we find a new equation for the parabola. By substituting the new rotated coordinates \( x = \frac{\sqrt{2}}{2} (X + Y) \) and \( y = \frac{\sqrt{2}}{2} (Y - X) \) into the original equation \( y = (x - 1)^2 \), we derive an expression in terms of \(X\) and \(Y\).

This transformation rewrites the parabola's equation into a new form, maintaining its geometric properties but oriented differently in the coordinate plane. This new representation simplifies the understanding and further manipulation of the conic in its new coordinate system.