What is the equation of an ellipse that is not aligned with the axis? I have the an ellipse with its semi-minor axis length $x$, and semi major axis $4x$.  However, it is oriented $45$ degrees from the axis (but is still centred at the origin).  I want to do some work with such a shape, but don't know how to express it algebraically.  What is the equation for this ellipse?

Sorry I didn’t see this one much earlier. Your ellipse has its major axis aligned with (I’m assuming) the line $y=x$, in other words it has symmetry $x\leftrightarrow y$. This means that it has an equation of the shape
$$
a(x^2+y^2) + bxy+c(x+y)=1\,,
$$
and since you’ve specified that the center is at the origin, you also need $c=0$. Thus your equation will simply be $a(x^2+y^2)+bxy=1$. You say that the semimajor axis is to be $4d$ and the semiminor axis is to be $d$ in length. So the points you know to be on the ellipse are $(\frac{4d}{\sqrt2},\frac{4d}{\sqrt2})=(2\sqrt2d,2\sqrt2d)$ at the end of the major axis and $(-\frac d{\sqrt2},\frac d{\sqrt2})$ on the minor axis.
Substituting these values into the equation and solving for $a$ and $b$, you get
$$
a=\frac{17}{32d^2}\,,\quad b=-\frac{15}{16d^2}\,,
$$
at least if I have not made a computational error. And there’s the equation for your ellipse.
