If the roots of the equation a(b - c)x2 + b(c - a)x + c(a - b) = 0 are

1.	If the roots of the equation a(b - c)x2 + b(c - a)x + c(a - b) = 0 are equal, show that 2/b=1/a+1/c.
Answer» Since the roots of the given equations are equal, so discriminant will be equal to zero. => b²(c - a)² - 4a(b - c) . c(a - b) = 0 => b²(c² + a² - 2ac) - 4ac(ba - ca - b² + bc) = 0, => a²b² + b²c² + 4a²c² + 2b²ac - 4ac²bc - 4abc² = 0 => (ab + bc - 2ac)²= 0 => ab + bc - 2ac = 0 => ab + bc = 2ac => 1/c+1/a=2/b => 2/b=1/a+1/c. Hence Proved.

If the roots of the equation a(b - c)x2 + b(c - a)x + c(a - b) = 0 are equal, show that 2/b=1/a+1/c.

Answer»

Since the roots of the given equations are equal, so discriminant will be equal to zero.

=> b²(c - a)² - 4a(b - c) . c(a - b) = 0

=> b²(c² + a² - 2ac) - 4ac(ba - ca - b² + bc) = 0,

=> a²b² + b²c² + 4a²c² + 2b²ac - 4ac²bc - 4abc² = 0

=> (ab + bc - 2ac)²= 0

=> ab + bc - 2ac = 0

=> ab + bc = 2ac

=> 1/c+1/a=2/b

=> 2/b=1/a+1/c.

Hence Proved.