Translation and Rotation of Coordinate

This section is about the basic operation for simplify the equation of graphs.

Translation of Coordinate

This is the basic method when handling coordinates. We can move the coordinate along x-axis or y-axis or both in order to simplify the equation.

A few methods are listed below.

Completing the Square

For second-degree equations in the form ${\displaystyle A{x}^2+C{y}^2+Dx+Ey+F=0}$, we can use this method to simplify the equation into the form ${\displaystyle (x+m{)}^2+(y+n{)}^2=k}$.

Rotation of Coordinate

For equations like ${\displaystyle A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0}$, we can not simply use translations to simplify the equation and make the coordinate fit the graph of the equation. We introduce the method of rotation here.

Just follow the steps we learn trig functions. We can use geometry for help to determine the relationship between ${\displaystyle (x,y)}$ and ${\displaystyle (u,v)}$. The detailed steps are given in page 526.

For a rotation angle ${\displaystyle \theta }$, we can find:

• ${\displaystyle x=u\cos \theta -v\sin \theta }$
• ${\displaystyle y=u\sin \theta +v\cos \theta }$

These two formulas determine the rotation of axes.

Find the Angle of ${\displaystyle \theta }$

A given formula about ${\displaystyle \theta }$ can eliminate the cross product of ${\displaystyle xy}$:

${\displaystyle \cot 2\theta =\frac{A-C}{B}}$