KVPY SA Exam 19-Nov-2017 Question Paper

Question : 64 of 80

Marks: +1, -0

Let

C_1, C_2

be two circles touching each other externally at the point A and let AB be the diameter of circle

C_1

.Draw a secant

BA_3

to circle

C_2

, intersecting circle

C_1

at a point

A_1

(≠A), and circle

C_2

at points

A_2

and

A_3 .

BA_1 = 2, BA_2 = 3

and

BA_3 = 4

, then the radii of circles

C_1

and

C_2

are respectively

Solution:

BM = A_1M = 1

A_1A_2 = 1

A_2N = A_3N =1/2

Let radius of

C_1 \is r_1

Let radius of

C_2 \is r_2

PM = √{r_1-1},QN=√{r_2^2-1/4}

∵

∆\QNB ~ ∆\PMB

∴

{√{r_2^2-1/4}}/{√{r_1-1}}

{BN}/{BM}={7\/2}/1

⇒

4r_2^2 = 49r^2_1-48

...(i)
Also, in ∆QNB

BQ^2 = BN^2 + NQ^2

(2r_1 + r_2 ) ^2 = 49/4+r_2^2-1/4

⇒

r ^2_1+ r_1 r_2 = 3

....(ii)
Solve (i) & (ii)

r_1=√{6/5}=√30/5\ &\ r_2={3√30}/10

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