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

Question

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_______

TestHub · Accepted Answer

​ 2 x + 3 y − 1 = 0 x + 2 y − 1 = 0 a x + b y − 1 = 0 ​ ​ ( 3 6 − 6 ​ , 3 8 − 8 ​ ) = ( 0 , 0 ) ​ ​ a x + b y − 1 = 0 ( − 1 − 0 1 − 0 ​ ) ( b − a ​ ) = − 1 ⇒ − a = b ⇒ ax − ay − 1 = 0 a x − a ( 1 − 3 2 x ​ ) − 1 x ( a + 3 2 a ​ ) = 3 a ​ x = 5 a a + 3 ​ 2 ( 5 a a + 3 ​ ) + 3 y − 1 = 0 y = 3 1 − 5 a 2 a + 6 ​ ​ = 3 × 5 a 3 a − 6 ​ y = 5 a a − 2 ​ ( 5 a a + 3 ​ ) ( 5 a a − 2 ​ ) ​ = 2 ⇒ a − 2 = 2 a + 6 a = − 8 b = 8 − 8 x + 8 y − 1 = 0 ∣ a − b ∣ = 16 ​

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