A circle with center $O$ has radius $8$ units and circle $P$ has radius $2$ units. The circles are externally tangent to each other at point $Q$. Segment $TS$ is the common external tangent to circle $O$ and circle $P$ at points $T$ and $S$, respectively. What is the length of segment $OS$? Express your answer in simplest radical form.

LeoIsTheBest Jun 22, 2019

#1**-6 **

DOUBLE POST FOUND: https://web2.0calc.com/questions/a-few-theoretically-short-problems

A circle with center O has radius 8 units and circle P has radius 2 units. The circles are externally tangent to each other at point Q. Segment TS is the common external tangent to circle O and circle P at points T and S, respectively. What is the length of segment OS? Express your answer in simplest radical form.

CuteDramione Jun 22, 2019

#3**+4 **

I think this problem is solved easily if we lay it out in this manner :

Let the tangent to the circles be the equation y = -8

Let the center of circle O be (0, 0)

Let the center of circle P be (a, - 6)

Let the equations of the two circles be :

x^2 + y^2 = 64

And

(x - a)^2 + (y - 6)^2 = 4

And ....since the distance between the centers of the circle is the sum of the radi, i,e., 10 units, we can find the x coordinate of the center of circle P, a, as follows :

(a - 0)^2 + (0 - 6)^2 = 10^2

a^2 + 36 = 100

a^2 = 64

a = 8

So...the center of P is (8, -6)

So S must have the coordinates (8, -8)

So....OS has a length of sqrt [ (8 - 0)^2 + (-8 - 0)^2 ] = sqrt [ 64 + 64] = sqrt (64 * 2) =

8 sqrt(2) units

CPhill Jun 25, 2019