Rotation of Axes
In Figure 1 the x- and y-axes have been rotated through an acute angle f about the origin to produce a new pair of axes, which we call the X- and Y-axes. A point P that has
coordinates 1x, y2 in the old system has coordinates 1X, Y2 in the new system. If we let
r denote the distance of P from the origin and let u be the angle that the segment OP
| 0 | |
| P(x, y) P(X, Y) X ƒ |
Y |
y
x
FiguRE 1
1.5 RotAtion oF AxES
In Section 1.4 we studied conics with equations of the form
Ax2 Cy2 Dx Ey F 0
makes with the new X-axis, then we can see from Figure 2 (by considering the two right
triangles in the figure) that
| X r cos u x r cos1u f2 |
Y r sin u y r sin1u f2 |
Using the Addition Formula for Cosine, we see that
x r cos1u f2
r1cos u cos f sin u sin f2
1r cos u2 cos f 1r sin u2 sin f
X cos f Y sin f
Similarly, we can apply the Addition Formula for Sine to the expression for y to obtain
RotAtion oF AxES FoRMulAS
Suppose the x- and y-axes in a coordinate plane are rotated through the acute
angle f to produce the X- and Y-axes, as shown in Figure 1. Then the coordinates 1x, y2 and 1X, Y2 of a point in the xy- and the XY-planes are related as
follows.
| x X cos f Y sin f y X sin f Y cos f |
X x cos f y sin f Y x sin f y cos f |
y
| 0 | x | |
| P X Y X |
Y | ¨ƒ y r |
x
FiguRE 2
CHAPTER 1 ■ Conic Sections
)
y X sin f Y cos f. By treating these equations for x and y as a system of linear
equations in the variables X and Y , we obtain expressions for X and Y
in terms of x and y, as detailed in the following box.
Dr.Mohana
