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Let C(1), C(2) be two circles touching ...

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)(neA),` and circle `C_(2)` at points `A_(2)` and `A_(3)` . If `BA_(1) = 2`, `BA_(2) = 3` and `BA_(3) = 4`, then the radii of circles `C_(1)` and `C_(2)` are respectively

A

`(sqrt30)/(5),(3sqrt30)/(10)`

B

`(sqrt5)/(2),(7sqrt5)/(10)`

C

`(sqrt6)/(2),(sqrt6)/(2)`

D

`(sqrt10)/(3),(17sqrt10)/(30)`

Text Solution

Verified by Experts

The correct Answer is:
A

`BM=A_(1)M=1`
`A_(1)A_(2)=1`
`"Let radius of "C_(1)" is "r_(1)`
`"Let radius of "C_(2)" is "r_(2)`
`PM=sqrt(r_(1)^(2)-1),QN=sqrt(r_(2)^(2)-(1)/(4))`
`becausetriangleQNB~trianglePMB`
`thereforesqrt(r_(2)^(2)-(1)/(4))/sqrt(r_(1)^(2)-1)=(BN)/(BM)=(7//2)/(1)`
`rArr 4r_(2)^(2)=49r_(1)^(2)-48...(i)`
`"Also, in" triangleQNB`
`BQ^(2)=BN^(2)+NQ^(2)`
`(2r_(1)+r_(2))^(2)=(49)/(4)+r_(2)^(2)-(1)/(4)`
`r_(1)^(2)+r_(1)r_(2)=3`
`"Solve(i) & (ii) "`
`r_(1)=sqrt((6)/(5))=(sqrt30)/(5)" & "r_(2)=(3sqrt30)/(10)`
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