Equal circles with centres O and O' touch each other at X. OO' produced to meet a circle with centre O , at A. AC is a tangent to the circle whose centre is O. O'D is perpendicular to AC. Find the value of DO'/CO.....cAn aNy1 hElP MeH WiD thIs qUeStIoN pLs...???????

 Hi parvathi ! :)

Here's how we did it. Hope you find it fine ! :D

And for the diagram of the question , check pg 10.31 of R.D Sharma.

Given : Two circles with centres O and O' are equal , which means that their respective radii are eual to each other. So , XO = XO'

2. Angle ADO'= 90 

Proof : Since that tangent passes through the circle with centre O at C , Angle OCD = 90 ( Radius perpendicular to tangent )

Now , we look at the two triangles in the figure , Triangle ADO' and triangle AOC.

In ADO' ---> AO' is the hypotenuse , as it is the side opp to the right angle.

Hence using the pythagoras theorem we can say that - AO' square = DO' square + AD square

So , DO' = root of (AD square - AO'square)

In Triangle AOC ---> By the pythagoras theorem , we can say that -  root of (AO square - ACsquare )= CO

Now we can take AO' as 'x' to make the steps ahead easier.

If AO' = x

then AO = 3x ( because the radii are equal , so AO = AO'+O'X+OX)

In the previous classes , we have learnt that "a line parallel to a side of a triangle intersects the other side in the same ratio".(pg 124 of tb)

Hence we can say that since A0'/O'D = x/2x = 1/2 ---> AD/DC= 1/2 and DC/2 = AD

Now , we substitute these values into the two pythagoras equations..

So , root of ( X square - DC square/4) = OD  -- 1.

And 9x square - (DC/2+ DC)square = OC square

-->root of ( 9x square - 9DC square/4 ) = OC --- 2.

Dividing  1. by 2. , we get 

Root of ( 4x square - DC square ) / Root of (36 DC square - 9DC square)

= Root of (I/9)

= 1/3