How do you rotate the axes to transform the equation #x^2+xy+y^2=2# into a new equation with no xy term and then find the angle of rotation?

Question

4135 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-02-25T10:25:47

See below.

#x^2+x y + y^2=2# can be read as

#(x,y)((1,1/2),(1/2,1))((x),(y))=(x,y)M((x),(y)) = 2#

Changing to a rotated axes

#((X),(Y))=((costheta,-sintheta),(sintheta,costheta))((x),(y)) = R((x),(y))# we have

#(X,Y)R^TMR((X),(Y)) = 2# where #R^TMR =((1+costhetasintheta,1/2cos(2theta)),(1/2cos(2theta),1-costheta sintheta)) #

Now if we choose #theta=pi/4# we have

#R^TMR =((1+1/2,0),(0,1-1/2))# and after the rotation the conic in the rotated axis reads as

#3/2X^2+1/2Y^2=2# which describes an ellipse.

NOTE: #R^T# is the transpose of #R# and #R# being orthonormal, #R^-1 = R^T#