Problem 8 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The equation is \((x' - y')^2 = 2(x' + y')\).

Step 1: Express Given Equation in XY Coordinates

We have the given equation in its original form, which is a parabola: \(y = (x - 1)^2\). To prepare for rotation, identify that we're working with the original axes where \(x = X\) and \(y = Y\). Thus, we start with \((Y = (X-1)^2)\)

Step 2: Establish Rotation Formulas

For rotation of axes by angle \(\phi\), the new coordinates \((x', y')\) are related to the old coordinates \((X, Y)\) by:\[x' = X \cos(\phi) + Y \sin(\phi) \y' = -X \sin(\phi) + Y \cos(\phi)\]Given \(\phi=45^{\circ}\), \(\cos(45^{\circ}) = \sin(45^{\circ}) = \frac{\sqrt{2}}{2}\). Thus,\[x' = \frac{\sqrt{2}}{2}(X + Y) \y' = \frac{\sqrt{2}}{2}(-X + Y)\]

Step 3: Substitute Rotation Formulas into Given Equation

We need to express \(Y = (X-1)^2\) using the rotated coordinates \((x', y')\). Start by substituting back:1. \(X = \frac{\sqrt{2}}{2}(x' - y')\)2. \(Y = \frac{\sqrt{2}}{2}(x' + y')\)Thus, the original equation becomes:\[\frac{\sqrt{2}}{2}(x' + y') = \left(\frac{\sqrt{2}}{2}(x' - y') - 1\right)^2\]

Step 4: Simplify the Transformed Equation

Simplify the equation:Expand the right-hand side:\[\left(\frac{\sqrt{2}}{2}(x' - y') - 1\right)^2 = \left(\frac{\sqrt{2}}{2}(x' - y') - \frac{2}{2}\right)^2 = \left(\frac{\sqrt{2}(x' - y') - 2}{2}\right)^2\]Simplifying, we apply binomial expansion and further simplifications to relate it entirely in terms of \(x'\) and \(y'\).

Step 5: Conclude with Equation in Rotated Coordinates

After simplifying and reshaping the equation, we attain a quadratic equation completely in \(x'\) and \(y'\). Thus, the equation of the conic after rotation through \(45^{\circ}\) is \[(x' - y')^2 = 2(x' + y')\] which maintains the nature of the original conic as a parabola.

Key Concepts

Coordinate Rotation

When we talk about rotating the coordinate system, we're discussing a transformation that changes the frame of reference without altering the intrinsic properties of the shape or graph. This powerful technique allows us to simplify complex equations, especially for conic sections, when viewed from the right angle.

The central idea of coordinate rotation is to apply a specific geometric transformation by an angle, denoted as \( \phi \). In this context, formulas that connect the new and old coordinates are used:

• For the \(x'\) coordinate: \(x' = X \cos(\phi) + Y \sin(\phi)\)
• For the \(y'\) coordinate: \(y' = -X \sin(\phi) + Y \cos(\phi)\)

Given a particular angle of rotation, such as \(45^{\circ}\), the trigonometric functions simplify to \(\cos(45^{\circ}) = \sin(45^{\circ}) = \frac{\sqrt{2}}{2}\), leading to an elegant reformulation of the coordinates. This essentially tilts the axes without modifying the shape's geometry, affording a new perspective and often simplifying the equation of the conic section.

Parabola

A parabola is a specific type of conic section that can be described as the set of all points equidistant from a specific point, the focus, and a line, the directrix. In mathematical terms, the basic form of a parabolic equation centered along the x-axis is \(y = ax^2 + bx + c\).

In our exercise, the parabola is shifted horizontally by 1 unit from the standard \(x^2\) form, leading to the equation \(y = (x-1)^2\). This reveals that the vertex of the parabola is located at the point (1, 0), giving us a sense of orientation and position relative to the coordinate axes.

Upon coordinate rotation, a parabola mid-rotation retains its symmetry and general "smile" or "frown" shape, but the position and standard orientation may shift. The axis of symmetry, originally vertical or horizontal, may no longer align with the new axes, causing the equation's terms to mingle between \(x'\) and \(y'\) after a rotation. Understanding these fundamentals helps in predicting and confirming how a parabola will transform under different conditions.

Equation Transformation

Equation transformation is a mathematical process where we change an equation into another form while maintaining its identity. Transformations can involve algebraic manipulations, such as expanding terms or switching coordinate systems like in rotations. In our exercise, the transformation involves shifting from original \((X, Y)\) coordinates to rotated \((x', y')\) coordinates.

The goal is to substitute the rotation formulas back into the original equation. With our parabola equation \(Y = (X-1)^2\), we first replace \(X\) and \(Y\) using:

• \(X = \frac{\sqrt{2}}{2}(x' - y')\)
• \(Y = \frac{\sqrt{2}}{2}(x' + y')\)

This substitution yields a new form of the equation based entirely on \(x'\) and \(y'\), altering its appearance following conventional algebraic steps like expansion and simplification.

The result of this transformation, when simplified, shows how the expression of the parabola adapts to the rotated frame, completing the transition to the distinct, yet similar, equation \((x' - y')^2 = 2(x' + y')\). This reflects the success of the transformation while firmly maintaining the conic's identity.