08_Ex 6.8

Exercise Questions

Questions	Links
Q1. Find an equation of each of the following curves…	6.8 Q-1
Q2. Find coordinates of the new origin so that…	6.8 Q-2
Q3. In each of the following, find an equation of…	6.8 Q-3
Q4. Find measure of the angle through which…	6.8 Q-4

Overview

This exercise introduces the concept of transforming conic equations by shifting the origin or rotating the axes. These transformations allow for simplifying the given equations and removing unwanted terms like the first-degree terms and the mixed product terms ( $xy$ ).

Applications of this technique include:

Simplifying the analysis of conic sections by eliminating linear and cross terms.
Making the equations of curves easier to interpret by re-aligning the coordinate system.

Key Concepts

Shifting the Origin: The transformation of conic equations by moving the origin to a new point involves substituting $x' = x - h$ and $y' = y - k$ , where $(h, k)$ is the new origin. This transformation removes first-degree terms when $h$ and $k$ are appropriately chosen.
Rotation of Axes: The rotation of axes through an angle $\theta$ changes the coordinates according to the formulas:

x' = x \cos\theta - y \sin\theta, \quad y' = x \sin\theta + y \cos\theta

The purpose of rotating the axes is to eliminate the product term $xy$ in the conic equation.

First-Degree Terms: When transforming conics by shifting the origin, choosing the right $h$ and $k$ values can eliminate first-degree terms in the transformed equation.

Important Formulas

Shifting the Origin:

x' = x - h, \quad y' = y - k

Rotation of Axes:

x' = x \cos\theta - y \sin\theta, \quad y' = x \sin\theta + y \cos\theta

For Transformation to Remove First-Degree Terms: The values of $h$ and $k$ are found by solving the system:

6h + 24 = 0, \quad 4k - 12 = 0

where these represent the conditions for removing the first-degree terms in the transformed equation.

Tips and Tricks

For shifting the origin:
- Focus on how the first-degree terms change when substituting $x' = x - h$ and $y' = y - k$ . Set these terms to zero to find the correct new origin.
For rotating the axes:
- If you’re dealing with a product term like $xy$ , aim to eliminate it by rotating the axes by the appropriate angle $\theta$ . Use the relationship $\tan 2\theta = \frac{B}{A - C}$ where $A, B, C$ are the coefficients from the general quadratic form.
Common mistakes to avoid:
- Don’t forget to simplify your equations after performing transformations.
- Check your steps carefully when solving for the new origin or angle, as errors in algebra can lead to incorrect transformed equations.

Summary

In this exercise, we focused on transforming conic equations by either shifting the origin or rotating the coordinate axes. These transformations simplify the equations, often by removing unwanted terms such as the first-degree terms or the mixed product term $xy$ . The main tools for solving these problems are the transformation equations for shifting the origin and rotating the axes, and the key to success lies in choosing the right values for $h, k,$ and $\theta$ to simplify the conic equations.