Question

if the equation (a^2 + b^2)x^2 - 2b (a+c) x + b^2 + c ^2 = 0 has equal roots then find the relation between a , b , and c

Answer 1

(a² + b²)x² -2b(a + c)x +( b² + c²) =0
have equal roots
so,
Discriminant =0
{2b( a + c )}² -4(a² + b²)(b² +c²) =0

b²( a² + c² +2ac ) - a²b² -a²c² -b⁴ -b²c² =0

b²a² + b²c² +2acb² -a²b² -a²c² -b⁴ -b²c² =0

-(a²c² + b⁴ -2ac.b²) =0

{(ac)² +(b²)² -2ac.b²} =0

(ac - b²)²=0

ac - b² =0

b² = ac

b/a = c/b

hence,
a, b, c are in Geometric progression .

Answer 2

D = 0 
b²-4ac = 0
(a² + b²)x² -2b(a + c)x +( b² + c²) =0
have equal roots 

so, 

D=0
b²-4ac = 0

{2b( a + c )}² -4(a² + b²)(b² +c²) =0
b²( a² + c² +2ac ) - a²b² -a²c² -b⁴ -b²c² =0
b²a² + b²c² +2acb² -a²b² -a²c² -b⁴ -b²c² =0
-(a²c² + b⁴ -2ac.b²) =0
{(ac)² +(b²)² -2ac.b²} =0
(ac - b²)²=0
ac - b² =0
b² = ac 
b/a = c/b