How do you use rotation of axes to identify and sketch the curve of $x^2+xy+y^2=1$ ?

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Answer 1 · 2016-11-14T11:46:51

An ellipse
$1/2(x_(pi/4))^2+3/2(y_(pi/4))^2=1$

This conic can be represented as

$p^TMp = 1$

with $p=(x,y)^T$ and $M = ((1,1/2),(1/2,1))$

Making a change of coordinates such that $q = (x_theta,y_theta)$

$q=R_(theta) p$ with $R_(theta)=((costheta,-sintheta),(sintheta,costheta))$ we have

$p = R_(theta)^Tq$ and

$p^TMp=q^TR_(theta)MR_(theta)^Tq = q^TM_(theta)q=1$

with $M_(theta) = ((1-sinthetacostheta,1/2cos2theta),(1/2cos2theta,1-sinthetacostheta))$

Choosing $theta$ such that $cos2theta=0$ or $2theta=pi/2$ or
$theta=pi/4$ we have

$M_(pi/4)=((1/2,0),(0,3/2))$ and in this new reference frame the conic looks as

$(x_(pi/4),y_(pi/4))M_(pi/4)((x_(pi/4)),(y_(pi/4))) = 1/2(x_(pi/4))^2+3/2(y_(pi/4))^2=1$

which is an ellipse.

How do you use rotation of axes to identify and sketch the curve of x^2+xy+y^2=1x^2+xy+y^2=1?