CAT 2004 | Question: 73

Question

A circle with radius $2$ is placed against a right angle. Another small circle is also placed as shown in the adjoining figure. What is the radius of the smaller circle?

• A. $3-2 \sqrt{2}$
• B. $4-2 \sqrt{2}$
• C. $7-4 \sqrt{2}$
• D. $6-4 \sqrt{2}$

Answer 1

If we draw a line joining the centres of the two circles and leading to the intersection point of the two tangents , that can help us arrive at the solution.

Let r be the radius of the smaller circle so distance between centre and the point of intersection of tangent = sqrt(r² + r²)   =  √2 r.

And similarly distance between centre of larger circle and the point of intersection of tangent  =  √ (2² + 2²)  =  2√2 r

Also this distance can be written as : radius of larger circle + radius of smaller circle + distance between centre of smaller circle to point of intersection of tangents              =    2 + r + √2 r.

Hence we have :

          2 + r + √2 r     =     2√2

==>    r(1 + √2)         =     2(√2 - 1)

==>    r                    =     2(√2 - 1) / (√2 + 1)

==>    r                    =   [  2(√2 - 1) / (√2 + 1)  ]  * [ (√2 - 1) / (√2 - 1) ]

                               =      2(√2 - 1)² / [ (√2)² -  (1)² ]

                               =      2(√2 - 1)² 

                               =      6 - 4√2

 

Hence 4) should be the correct answer.