Finding a New Representation of the Given Equation after Rotating through a Given Angle
Until now, we have looked at equations of conic sections without an xy term, which aligns the graphs with the x- and y-axes. When we add an xy term, we are rotating the conic about the origin. If the x- and y-axes are rotated through an angle, say θ, then every point on the plane may be thought of as having two representations: (x,y) on the Cartesian plane with the original x-axis and y-axis, and (x′,y′) on the new plane defined by the new, rotated axes, called the x'-axis and y'-axis.
Figure 3. The graph of the rotated ellipse x2+y2−xy−15=0
We will find the relationships between x and y on the Cartesian plane with x′ and y′ on the new rotated plane.
Figure 4. The Cartesian plane with x- and y-axes and the resulting x′− and y′−axes formed by a rotation by an angle θ.
The original coordinate x- and y-axes have unit vectors i and j. The rotated coordinate axes have unit vectors i′ and j′. The angle θ is known as the angle of rotation. We may write the new unit vectors in terms of the original ones.
i′=cosθi+sinθjj′=−sinθi+cosθj
Figure 5. Relationship between the old and new coordinate planes.
Consider a vectoruin the new coordinate plane. It may be represented in terms of its coordinate axes.
u=x′i′+y′j′u=x′(icosθ+jsinθ)+y′(−isinθ+jcosθ)u=ix’cosθ+jx’sinθ−iy’sinθ+jy’cosθu=ix’cosθ−iy’sinθ+jx’sinθ+jy’cosθu=(x’cosθ−y’sinθ)i+(x’sinθ+y’cosθ)jSubstitute.Distribute.Apply commutative property.Factor by grouping.
Because u=x′i′+y′j′, we have representations of x and y in terms of the new coordinate system.
x=x′cosθ−y′sinθandy=x′sinθ+y′cosθ
A General Note: Equations of Rotation
If a point (x,y) on the Cartesian plane is represented on a new coordinate plane where the axes of rotation are formed by rotating an angle θ from the positive x-axis, then the coordinates of the point with respect to the new axes are (x′,y′). We can use the following equations of rotation to define the relationship between (x,y) and (x′,y′):
x=x′cosθ−y′sinθ
and
y=x′sinθ+y′cosθ
How To: Given the equation of a conic, find a new representation after rotating through an angle.
Find x and y where x=x′cosθ−y′sinθ and y=x′sinθ+y′cosθ.
Substitute the expression for x and y into in the given equation, then simplify.
Write the equations with x′ and y′ in standard form.
Example 2: Finding a New Representation of an Equation after Rotating through a Given Angle
Find a new representation of the equation 2x2−xy+2y2−30=0 after rotating through an angle of θ=45∘.
Solution
Find x and y, where x=x′cosθ−y′sinθ and y=x′sinθ+y′cosθ.
Because θ=45∘,
)2)2(x′−y′)(x′−y′)−2(x′−y′)(x′+y′)+)2)2(x′+y′)(x′+y′)−30=0x′2)−2x′y′+y′2−2(x′2−y′2)+x′2)+2x′y′+y′2−30=02x′2+2y′2−2(x′2−y′2)=302(2x′2+2y′2−2(x′2−y′2))=2(30)4x′2+4y′2−(x′2−y′2)=604x′2+4y′2−x′2+y′2=60603x′2+605y′2=6060FOIL methodCombine like terms.Combine like terms.Multiply both sides by 2.Simplify.Distribute.Set equal to 1.
Write the equations with x′ and y′ in the standard form.
20x′2+12y′2=1
This equation is an ellipse. Figure 6 shows the graph.