Ow it's just a line family! Right? Yes
Consider a family of circles passing through the intersection points of the lines and and having its centre on the acute angle bisector of the given lines.
(a)Show that the common chords of each member of the family and the circle are concurrent.
(b)If the point of concurrency is (a,b) then the value of a+b is ?
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1 solution
Solve the lines to get the intersection point as P(1,1).The acute angle bisector is the line y=x.Let centre of a variable circle C be O (k,k) .The equation of C can be taken to be (k,k).
C: ( k − 1 ) 2 + ( k − 1 ) 2 = ( x − k ) 2 + ( y − k ) 2 . This gives the equation of C.
The common chord of C with the given circle S (say) is given by C-S=0
We get ( 4 x − 6 y + 7 ) + 2 k ( x + y − 2 ) = 0 This is a family of lines.Solve 4 x − 6 y + 7 = 0 and x + y − 2 = 0 to get the intersection as ( 1 / 2 , 3 / 2 )