# Find ellipse axis knowing two tangents and orientation of axis

I've been looking at this for hours now, trying to solve it using the ellipse equation and looking at other questions here, unfortunately with no success.

Here is what I am trying to achieve: I have:

• big axis of ellipse is parallel to x axis
• ellipse is tangent to x axis in origin (= point A)
• the position of point D
• the slope of the tangent to the ellipse in D

I am looking for:

• length of major axis of the ellipse (EF)
• length of minor axis of the ellipse (AG)

How would you approach this? I feel like there should be only one solution to this problem and that it is well defined, but I could not solve it...

Thanks a lot for your help

Let $$D(x_d,y_d)$$ and $$k$$ the slope of the tangent. The ellipse has the form,

$$\frac{x^2}{a^2} + \frac{(y-b)^2}{b^2} = 1$$

Its derivative is $$y'=-\frac{b^2x}{a^2(y-b)}$$ and matches $$k$$ at D,

$$k = -\frac{b^2x_d}{a^2(y_d-b)} \tag{1}$$

The point $$(x_d,y_d)$$ satisfies,

$$\frac{x_d^2}{a^2} + \frac{(y_d-b)^2}{b^2} = 1\tag{2}$$

Combine (1) an (2) to get the following equation for $$b$$,

$$(y_d-b)^2-kx_d(y_d-b)=b^2$$

which yields the unique solution for the minor axis $$b$$,

$$b= \frac{y_d(y_d-kx_d)}{2y_d-kx_d}$$

Then, plug the solution $$b$$ into (1) to get the major axis $$a$$.

• It might be worth noting that besides the obvious condition $2d_y-kx_d=0$, there’s also no such ellipse when $b\le0$. – amd Oct 9 '19 at 20:13
• Thanks, I was missing multiple steps indeed! – zigoingoin Oct 10 '19 at 18:50