Let circle C be the image of x 2 + y 2 − 2 x + 4 y − 4 = 0 in the line 2 x − 3 y + 5 = 0 and A be the point on C such that O A is parallel to x -axis and A lies on the right hand side of the centre O

Question

Let circle C be the image of x 2 + y 2 − 2 x + 4 y − 4 = 0 in the line 2 x − 3 y + 5 = 0 and A be the point on C such that O A is parallel to x -axis and A lies on the right hand side of the centre O of C . If B ( α , β ) , with β < 4 , lies on C such that the length of the are A B is ( 1/6 ) th of the perimeter of C , then β − 3 ​ α is equal to

TestHub · Accepted Answer

Centre ( 1 , − 2 ) , r = 3 Reflection of ( 1 , − 2 ) about 2 x − 3 y + 5 = 0 ​ 2 x − 1 ​ = − 3 y + 2 ​ = 13 − 2 ( 2 + 6 + 5 ) ​ = − 2 x = − 3 , y = 4 ​ Equation of circle ' C ' C : ( x + 3 ) 2 + ( y − 4 ) 2 = 9 A.T.Q. ​ ℓ ( arcAB ) = 6 1 ​ × 2 π r r θ = 6 1 ​ × 2 π r θ = 3 π ​ ( α + 6 ) 2 + ( β − 4 ) 2 = 27 ( α + 6 ) 2 − ( α + 3 ) 2 = 18 ( α + 3 ) 2 ± ( β − 4 ) 2 = 9 ​ ⇒ 6 α = − 9 ​ ​ ⇒ α = 2 − 3 ​ , β = ( 4 − 2 3 3 ​ ​ ) ∴ β − 3 ​ α ( 4 − 2 3 3 ​ ​ ) + 2 3 3 ​ ​ = 4 ​

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