Question

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)`

Answer 1

Correct Answer - 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)`