Translation and Rotation of Axes

The following examples demonstrate how to apply the formulas and geometric properties of translation and rotation of axes to solve analytical geometry problems, such as finding coordinates in a new system and eliminating specific terms from the general conic equation.

Translation of Axes

Example 1: Translating a Point to a New Coordinate System

Problem: A point PP has coordinates (5,−3)(5, -3) in the original (x,y)(x, y) coordinate system. If the axes are translated such that the new origin is situated at the point (2,−1)(2, -1), what are the new coordinates (x′,y′)(x', y') of point PP?

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Example 2: Translating an Equation to Eliminate Linear Terms

Problem: Translate the axes to eliminate the first-degree (linear) terms from the equation x2+y2−4x+6y−3=0x^2 + y^2 - 4x + 6y - 3 = 0. State the coordinates of the new origin and the simplified equation in the translated system.

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Rotation of Axes

Example 3: Determining the Angle of Rotation to Eliminate the xy-Term

Problem: Find the angle of rotation required to eliminate the xyxy-term from the conic equation 5x2−4xy+8y2−36=05x^2 - 4xy + 8y^2 - 36 = 0.

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Example 4: Transforming a Conic Equation Using Rotation of Axes

Problem: Given the equation xy=2xy = 2, rotate the axes by an angle θ=45∘\theta = 45^{\circ} to find the new equation in terms of x′x' and y′y'. Identify the type of conic section.

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