Pair of Circles (Non intersecting)

Question

I have Two circles (At present moment, non-intersecting). Circle 1: Centre (xc1,yc1); Radius R1. Circle 2: Centre (xc2,yc2); Radius R2. An arbitrary point P (xp,yp) which will lie anywhere in the plane except inside circles. Theta1 is the angle made by the positive x-axis from centre of circle 1 to point P (xp,yp). Theta2 is the angle made by the positive x-axis from centre of circle 2 to point P.

How to find relation between angles theta1 and theta2?

C1(xc1,yc1), R1, C2(xc2,yc2), R2, P(xp,yp) are all known.

Thank you.

A quick glance suggests that the angle measured at $P$ is always the difference between $\theta_1$ and $\theta_2$. What sort of relation between them are you interested in? — abiessu, Jun 30 '22 at 16:07

Narasimham · Answer 1 · 2022-06-30T18:41:25.360

1

HINT:

By the Law of Sines in a triangle,

$$\frac{R_2}{\sqrt{(xc_2^2- xc_1^2)+(yc_2^2- yc_1^2)}}< \frac {\sin\theta_1}{\sin(\theta_2-\theta_1)} $$

If P is on the boundary of the second circle we have an equality.

edited Jun 30 '22 at 18:41

answered Jun 30 '22 at 18:36

Narasimham

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Pair of Circles (Non intersecting)

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