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If a,b,c are distinct and the roots of (b-c) x2 + (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
Correct answer is option 'A'. Can you explain this answer?
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If a,b,c are distinct and the roots of (b-c)x2+ (c-a) x + (a-b) = 0 ar...
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If a,b,c are distinct and the roots of (b-c)x2+ (c-a) x + (a-b) = 0 ar...
Explanation:
Given:
a,b,c are distinct roots of the quadratic equation (b-c)x^2 + (c-a)x + (a-b) = 0
Equal Roots:
When the roots of a quadratic equation are equal, the discriminant of the equation is equal to zero.
In this case, the discriminant is:
(c-a)^2 - 4(b-c)(a-b) = 0
c^2 - 2ac + a^2 - 4ab + 4bc = 0
Solving for a, b, c:
By simplifying the above equation, we get:
c^2 - 2ac + a^2 - 4ab + 4bc = 0
=> a^2 - 2ac + c^2 = 4ab - 4bc
=> a^2 - 2ac + c^2 = 4a(b - c)
This equation shows that a, b, and c are in an arithmetic progression.
Therefore, the correct answer is option 'A' - Arithmetic progression.
Given:
a,b,c are distinct roots of the quadratic equation (b-c)x^2 + (c-a)x + (a-b) = 0
Equal Roots:
When the roots of a quadratic equation are equal, the discriminant of the equation is equal to zero.
In this case, the discriminant is:
(c-a)^2 - 4(b-c)(a-b) = 0
c^2 - 2ac + a^2 - 4ab + 4bc = 0
Solving for a, b, c:
By simplifying the above equation, we get:
c^2 - 2ac + a^2 - 4ab + 4bc = 0
=> a^2 - 2ac + c^2 = 4ab - 4bc
=> a^2 - 2ac + c^2 = 4a(b - c)
This equation shows that a, b, and c are in an arithmetic progression.
Therefore, the correct answer is option 'A' - Arithmetic progression.
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