Ellipse from two arbitrary points, tangent at P1 and a focal point

Question

Is it possible to find this? Really only need the semi major axis or even it's orientation.

Please see the linked image. Known elements are in red and the desired element is in blue.

Answer 1

It’s possible, but it’s not going to be pretty.

One approach is to find the second focus. The three known points determine an hyperbola along which this point lies—see this question for details.

In terms of these two foci, a formula for the slope of the tangent to the ellipse can be found via implicit differentiation. Setting the known focus at the origin and denoting the other focus by $F$, we end up with the expression $$-{\frac{x_{P_1}}{\|P_1\|}+\frac{x_{P_1}-x_F}{\|P_1-F\|}\over\frac{y_{P_1}}{\|P_1\|}+\frac{y_{P_1}-y_F}{\|P_1-F\|}}$$ for the slope of the tangent at $P_1$. This gives you another equation for $F$, which should be enough to determine the location of the second focus. Once you have that, the semimajor axis length is easily recovered—the sum of the distances of a point on the ellipse to the two foci is equal to $2a$.