$x^2 + xy + y^2 = 7$
$x$-axis = $\frac{\sqrt{21}}{3}$, $\frac{-2\sqrt{21}}{3}$
I don't understand how to find the $y$-axis.
$x^2 + xy + y^2 = 7$
$x$-axis = $\frac{\sqrt{21}}{3}$, $\frac{-2\sqrt{21}}{3}$
I don't understand how to find the $y$-axis.
Usually for these problems one uses the implicit derivative, which for this problem is $y'=\frac{-2x-y}{x+2y}.$ Then horizontal tangents occur when the top is zero, i.e. $2x+y=0$, and vertical tangents occur when the bottom is zero, i.e. $x+2y=0$. These equations are then plugged into the original relation $x^2+xy+y^2=7$ to get the actual coordinates of the points.
Note: Just noticed that Joe Johnson made this same suggestion re. implicit derivative!
EDIT: I got the horizontal tangents occur at $(x,y)=(+\sqrt{7/3},-2\sqrt{7/3})$ and at $(x,y)=(-\sqrt{7/3},+2\sqrt{7/3}).$ The vertical tangent points were like these, only switch the ordering of the pairs $(x,y)$. Maybe because the original ellipse $x^2+xy+y^2=7$ has its major axis at 45 degrees rotated.