Question

If the roots of the equation (a - b)x2 + (b-cbx + (c-a) = 0 are equal, show that c, a and
b are in AP​

Answer 1

Answer:

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

$d = \sqrt{ {b}^{2} - 4ac } = 0 \\ {(b - c)}^{2} = 4(a - b)(c - a) \\ {b}^{2} + {c}^{2} - 2bc = \\ 4ac - 4 {a}^{2} - 4bc + 4ba$

${b}^{2} + {c}^{2} + 4 {a}^{2} = 4ac + 4ba - 2bc$

So,

$2a = b + c$

So, b, a and c are in A. P.