If the orthocentre of the triangle formed by the lines 2x+3y−1=0, x+2y−1=0 and ax+by−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 ______.
Answer: 16
Solution
The centroid of the second triangle divides its circumcentre(3,4)–orthocentre(−6,−8) line so that G=(32(3)+1(−6),32(4)+1(−8))=(0,0). This G is the orthocentre of the triangle formed by 2x+3y−1=0, x+2y−1=0, ax+by−1=0. The first two meet at (−1,1); the altitude condition through (0,0) perpendicular to the side gives −a=b, and forcing the orthocentre at the origin yields a=−8,b=8. Hence ∣a−b∣=16.
