If the orthocentre of the triangle formed by the lines 2 x+3 y-1=0, x+2 y-1=0 and a x+b y-1=0 , is the centroid of another triangle, whose circumcentre and orthocentre respectively are (3,4) and (-6,-8) , then the value of |a-b| is_______

Step-by-step solution

aligned & 2 x+3 y-1=0 & x+2 y-1=0 & a x+b y-1=0 aligned aligned & ( 6-6 3 , 8-8 3 ) & =(0,0) aligned aligned & a x+b y-1=0 & ( 1-0 -1-0 ) ( -a b )=-1 & - a = b & ax - ay -1=0 & a x-a (1- 2 x 3 )-1 & x (a+ 2 a 3 )= a 3 & x = a +3 5 a & 2 ( a+3 5 a )+3 y-1=0 & y= 1- 2 a+6 5 a 3 = 3 a-6 3 5 a & y= a-2 5 a & ( a-2 5 a ) ( a+3 5 a ) =2 a-2=2 a+6 & a=-8 & ~b =8 & -8 x +8 y -1=0 & |a-b|=16 & aligned