# Length of line segment intersecting concentric circles

I am working on developing a model of how the intensity of solar rays change throughout a solar day as a function of the distance travelled through the atmosphere. That distance increases as the sun proceeds from its highest point to sunset. Another way to say this would be that the sun's rays are parallel throughout the day and enter the atmosphere at increasingly oblique angles as the afternoon progresses.

Given:

• two circles with centers at point $$A (0,0)$$ with radii $$r_1$$ and $$r_2$$ such that $$r_1
• line $$l$$ parallel to the $$x$$ axis and intersecting both circles
• a known angle theta formed between $$r_1$$ and the $$x$$-axis

Determine the distance between the two points at which line $$l$$ intersects the two circles.

I wish I had enough creds to include a drawing, but I hope someone will take enough interest to sketch this out for themselves. Thanks!

• hello,welcome to Mathematics Stack Exchange,Please provide your attempt to let the community know where exactly you are stuck Sep 15, 2023 at 3:41
• So you want something like the figure, and find the length of $ED$? Sep 15, 2023 at 4:34
• Thank you Andrei for the sketch. Yes this is what I am after. Sep 15, 2023 at 12:42

The coordinates of $$E$$ are $$(r_1\cos\theta,r_1\sin\theta)$$, coordinates of $$D$$ are $$(r_2\cos\phi,r_2\sin\phi)$$, and $$r_1\sin\theta=r_2\sin\phi$$, and you want $$r_2\cos\phi-r_1\cos\theta$$. Now, $$\sin\phi=r_1r_2^{-1}\sin\theta$$, so $$\cos\phi=\sqrt{1-(r_1r_2^{-1}\sin\theta)^2}$$, so your answer is $$\sqrt{r_2^2-(r_1\sin\theta)^2}-r_1\cos\theta$$.