How does $x^2+y^2+xy=\frac12$ become the ellipse $\frac{X^2}{1/3}+\frac{Y^2}{1}=1$?

Question

In the problem, I need to find the curve of the intersection of the plane $x+y+z=1$ and the sphere $x^2+y^2+z^2=1$.

I can visualise that the intersection would be an ellipse and when I substitute on equation into the other, I obtain $x^2+y^2+xy=1/2$. However, I don't understand what this equation represents since it's not equation for an ellipse.

In the solutions, $x^2+y^2+xy=1/2$ becomes the ellipse $\dfrac{X^2}{1/3}+\dfrac{Y^2}{1}=1$, which I'm having trouble understanding where it comes from.

What does $x^2+y^2+xy=1/2$ represent? and how we find the ellipse equation from that?

Any help is very appreciated. Thanks!

Does this answer your question? How does this equation $x^2+y^2+xy=1/2$ represent an ellipse? - found using an Approach0 search. Note that your $X^2/(1/3)+Y^2/(1)=1$ is considered the "standard" form, with this shown in answer #1 & answer #2 of this proposed duplicate question. — John Omielan, Apr 18 '23 at 20:11
Essentially you take the equation of the rotated ellipse and reconstruct the original ellipse, accounting for long axis to be horizontal and finding a principal angle (positive less than $2\pi$) $\theta$. The angle solves by forcing the coefficient of xy to be zero.
$$\begin{align}x_r^2+y_r^2+x_ry_r=2\ \Leftrightarrow (x\cos\theta-y\sin\theta)^2+(x\sin\theta+y\cos\theta)^2+ (x\cos\theta-y\sin\theta) (x\sin\theta+y\cos\theta)=2\ \Leftrightarrow \frac {x^2}{\frac 13}+\frac{y^2}1=1\end{align}$$. There is also a formula for calculating the angle straight off coefficients. — WindSoul, Apr 18 '23 at 20:24
There’s a mistake in the comment above: the first two equations the right side should be $\frac 12$, mistakenly is 2. — WindSoul, Apr 18 '23 at 20:54
An intersection of a plane and a sphere -- if exists -- is always a circle. Probably you wrongly set the problem. — user, Apr 18 '23 at 21:07
Please do not use tags that are irrelevant and inappropriate. — Ted Shifrin, Apr 18 '23 at 23:37
The curve is a circle. Do you really want to project it and get an ellipse? — Ted Shifrin, Apr 18 '23 at 23:38
Geogebra3d parametrizes the intersection $\langle x+y+z-1, x^2+y^2+z^2-1\rangle$ as the circle $(\frac13-\frac1{\sqrt3}\cos{t}+\frac13\sin{t},\frac13+\frac1{\sqrt3}\cos{t}+\frac13\sin{t},\frac13-\frac23\sin{t}).$ — Jan-Magnus Økland, Apr 19 '23 at 06:14

How does $x^2+y^2+xy=\frac12$ become the ellipse $\frac{X^2}{1/3}+\frac{Y^2}{1}=1$?

0 Answers0