If the orthocentre of the triangle, whose vertices are 1 , 2 , 2 , 3 and 3 , 1 is α , β , then the quadratic equation whose roots are α + 4 β and 4 α + β , is

Step-by-step solution

We have, Here m B H × m A C = - 1 ⇒ β - 3 α - 2 1 - 2 = - 1 ⇒ β - 3 = 2 α - 4 ⇒ β = 2 α - 1 ....(1) m A H × m B C = - 1 ⇒ β - 2 α - 1 - 2 = - 1 ⇒ 2 β - 4 = α - 1 Using (1), we can write: ⇒ 2 ( 2 α - 1 ) - 4 = α - 1 ⇒ 2 2 α - 1 = α + 3 ⇒ 3 α = 5 α = 5 3 ,   β = 7 3 ⇒ H 5 3 , 7 3 α + 4 β = 5 3 + 28 3 = 33 3 = 11 β + 4 α = 7 3 + 20 3 = 27 3 = 9 Re