A square is inscribed in the circle x 2 + y 2 − 6 x + 8 y − 103 = 0 with its sides parallel to the coordinate axes. Then the distance of the vertex of this square which is nearest to the origin is:

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The equation of circle is, x 2 + y 2 − 6 x + 8 y − 103 = 0 ⇒ ( x − 3 ) 2 + ( y + 4 ) 2 = ( 8 2 ​ ) 2 C ( 3 , − 4 ) , r = 8 2 ​ ⇒ Length of side of square = 2 ​ r = 16 ⇒ P ( − 5 , 4 ) , Q ( − 5 , − 12 ) R ( 11 , − 12 ) , S ( 11 , 4 ) ⇒ Required distance = OP = ( − 5 − 0 ) 2 + ( − 4 − 0 ) 2 ​ = 25 + 16 ​ = 41 ​

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