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Two circles $(C_1$ with radius $r_1$, and $C_2$ with radius $r_2$) are having the same centre. $P_1$ and $P_2$ are two distinct points lying on the circumference of $C_1$. The length of line $P_1P_2$ is $l$ units and it is tangent to $C_2$.

How can I express $r_2$ in terms of $r_1$ and $l$?

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If $P_1P_2$ is tangent to the innermost circle, then the distance from the center to $P_1P_2$ is $r_2.$ You know that the distance from the center to either $P_1$ or $P_2$ is $r_1.$ You only need to use the Pythagorean theorem to find the value you need.

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    $\begingroup$ Then, $r_2=\frac{1}{2}\sqrt{4r_1^2-l^2}$. Am I right? $\endgroup$ – Hussain-Alqatari Oct 30 '18 at 9:43
  • $\begingroup$ Precisely, @Hussain-Alqatari $\endgroup$ – Patricio Oct 30 '18 at 10:14

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