Question

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Note ⤵️

=> I don't know the answer
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=> Want proper logic

Answer 1

Final Answer : MN = 18cm


For Calculation see pic

Answer 2

first we find radius of big circle let it be R
radius of small circlebe r
let centre of concentric circles be O
draw a perpendicularto chord AB
it meet chord midpoint E
consider two right angled triangles
OED and OEB
sinO=12/r
sinO=15/R
12/r=15/R
4/r=5/R
R/r=5/4
coming to tangent MN
draw a perpendicular to MN
lenth of MNbe L
perpendicular meet MN at P
let us consider right angled triangleOPM
cosO=r/R=4/5
tanO=3/4=L/2r
L=3/2r
as a right angled triangle cosO=4/5
may be mn=6 cm