The equation `(a^(2)+b^(2))x^(2)-2b(a+c)x+(b^(2)+c^(2))=0` has equal roots. Which one of the following is correct about a,b and c ?

Options

A. They are in AP
B. They are in GP
C. They are in HP
D. They are neither in AP, nor in GP, nor in HP

Answer

B

Explanation

The given equation
`(a^(2)+b^(2))x^(2)-2b(a+c)x+(b^(2)+c^(2))=0`
has equal roots, so, discriminant = 0
Hence, `{2b(a+c)}^(2)-4(a^(2)+b^(2))(b^(2)+c^(2))=0`
`implies" "4b^(2)(a^(2)+c^(2)+2ca)-4(a^(2)b^(2)+a^(2)c^(2)+b^(4)+b^(2)c^(2))`
`=0`
`implies" "b^(2)a^(2)+b^(2)c^(2)+2b^(2)ca-a^(2)b^(2)-a^(2)c^(2)-b^(4)-b^(2)c^(2)=0`
`implies" "2b^(2)ca=b^(4)+a^(2)c^(2)`
`implies" "b^(4)-2b^(2)ca+a^(2)c^(2)=0`
`implies" "(b^(2))^(2)-2(b^(2))(ac)+(ac)^(2)=0`
`implies" "(b^(2)-ac)^(2)=0`
`implies" "b^(2)=ac`
`implies" "a,b,c" are in GP."`