Let C 1 be the circle in the third quadrant of radius 3 , that touches both coordinate axes. Let C 2 be the circle with centre ( 1 , 3 ) that touches C 1 externally at the point ( α , β ) . If (

Question

Let C 1 ​ be the circle in the third quadrant of radius 3 , that touches both coordinate axes. Let C 2 ​ be the circle with centre ( 1 , 3 ) that touches C 1 ​ externally at the point ( α , β ) . If ( β − α ) 2 = n m ​ , gcd ( m , n ) = 1 , then m + n is equal to :

TestHub · Accepted Answer

C 1 ​ : ( x + 3 ) 2 + ( y + 3 ) 2 = 3 2 Let C 1 ​ and C 2 ​ has centres A ( − 3 1 ​ − 3 ) and B ( 1 , 3 ) ​ AB = 16 + 36 ​ = 2 13 ​ r 1 ​ = 3 and r 2 ​ = 2 13 ​ − 3 P ( α , β ) , α = r 1 ​ + r 2 ​ r 1 ​ ( 1 ) + r 2 ​ ( − 3 ) ​ , β = r 1 ​ + r 2 ​ r 1 ​ ( 3 ) + r 2 ​ ( − 3 ) ​ α = 2 13 ​ 3 − 3 ( 2 13 ​ − 3 ) ​ , β = 2 13 ​ 18 − 6 13 ​ ​ ( β − α ) 2 = ( 2 13 ​ 6 ​ ) 2 ( β − α ) 2 = ( 2 13 ​ 6 ​ ) 2 , m + n = 22 ​

Mathematics - Circle Question with Solution | TestHub

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