How to draw a ellipse which passes two given points and is tangent to two given line?

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I was trying to draw Dandelin spheres using an ellipse. Then, I had to construct an ellipse that is tangent to two lines and passes two given points on the angle bisector of the two lines.

I attempted to draw a two-line tangent ellipse and dilate it to the points given.

But it didn't work, like below:

enter image description here

Now, how should I deal with it..?

BTW, this is my current progress with drawing the Dandelin spheres.

enter image description here