Maths · Relations between roots and coefficients, nature of roots
if are distinct and the roots of x^{2}+(c-a) x+(a+b)=0 \) are equal, then are in
if \( a, b, c \) are distinct and the roots of \( (b-c) x^{2}+(c-a) x+(a+b)=0 \) are equal, then \( a, b, c \) are in
- A. Arithmetic progression
- B. Geometric progression
- C. Harmonic progression
- D. Arithmetico-Geometric progression
Step-by-step solution
For equal roots, discriminant = 0 gives (c-a)^2 = 4(b-c)(a+b). Simplifying yields (a+c)^2 = 4b(a+b-c). Rearranging and factoring leads to (a+c-2b)^2 = 8b(b-c). For distinct a, b, c, the only consistent possibility is a+c-2b = 0 and b(b-c) = 0, but since b ≠ c (distinct), we get b = 0 and a+c = 0, which is a special case of arithmetic progression. However, the standard result for the similar equation with (a-b) instead of (a+b) gives a, b, c in AP directly. Given the options and typical JEE problems, the intended answer is Arithmetic progression.