In the below figure, two equal circles, with centres O and O', touch each other at X. OO' produced meets the circle with centre O' at A. AC is tangent to the circle with centre O, at the point C. O'D is perpendicular to AC. Find the value of

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Solution

Q.
In the below figure, two equal circles, with centres O and O', touch each other at X. OO' produced meets the circle with centre O' at A. AC is tangent to the circle with centre O, at the point C. O'D is perpendicular to AC. Find the value of $\frac{D O^{'}}{C O}$ .

Ans.
(1/3)

AC is tangent to circle with centre O.

Thus, $∠ A C O = 90 °$

In $∆ A O' D$ and $∆ A O C$ , $∠ A D O' = ∠ A C O = 90 °$

$∠ A = ∠ A$ (Common)

$∴$ $∆ A O' D ~ ∆ A O C (By AA similarity)$

$\Rightarrow \frac{A O'}{A O} = \frac{D O'}{C O}$

Now, $A O = A O' + O' X + X O = 3 r$

$∴$ $\frac{D O'}{C O} = \frac{r}{3 r} = \frac{1}{3}$

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