Exams Joint Entrance Examination Mathematics Circle Question 219 Question 219 of 219 Medium MCQ
Let C 1 C_1 C 1 be the circle in the third quadrant of radius 3 , that touches both coordinate axes. Let C 2 C_2 C 2 be the circle with centre ( 1 , 3 ) (1,3) ( 1 , 3 ) that touches C 1 \mathrm{C}_1 C 1 externally at the point ( α , β ) (\alpha, \beta) ( α , β ) . If ( β − α ) 2 = m n (\beta-\alpha)^2=\frac{m}{n} ( β − α ) 2 = n m , gcd ( m , n ) = 1 \operatorname{gcd}(m, n)=1 gcd ( m , n ) = 1 , then m + n m+n m + n is equal to
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16 + 36 = r + 3 ⇒ r = 52 − 3 ⇒ 3 − 3 r − r + 3 = α , β = 9 − 3 r 3 + r ⇒ ( 9 − 3 r − 3 + 3 r ) 2 ( r + 3 ) 2 ⇒ 36 12 = 9 13 = m n ⇒ m + n = 22 \begin{aligned}
& \sqrt{16+36}=r+3 \\
& \Rightarrow r=\sqrt{52}-3 \\
& \Rightarrow \frac{3-3 r}{-r+3}=\alpha, \beta=\frac{9-3 r}{3+r} \\
& \Rightarrow \frac{(9-3 r-3+3 r)^2}{(r+3)^2} \\
& \Rightarrow \frac{36}{12}=\frac{9}{13}=\frac{m}{n} \Rightarrow m+n=22
\end{aligned} 16 + 36 = r + 3 ⇒ r = 52 − 3 ⇒ − r + 3 3 − 3 r = α , β = 3 + r 9 − 3 r ⇒ ( r + 3 ) 2 ( 9 − 3 r − 3 + 3 r ) 2 ⇒ 12 36 = 13 9 = n m ⇒ m + n = 22 \] Answer the question to unlock