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

4358 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, $R^-1 = R^T$

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