Suppose a, b>0 and x1, x2, x3(x1>x2>x3) are roots of (x-a)/(b)+(x-b)/(a)=(b)/(x-a)+(a)/(x-b) and

step 1 So in this question, x minus a by b plus x minus b, x minus b, x minus b by a is equal to b by x

Suppose a, b>0 and x1, x2, x3(x1>x2>x3) are roots of...

Suppose $a, b>0$ and $x_{1}, x_{2}, x_{3}\left(x_{1}>x_{2}>x_{3}\right)$ are roots of $\frac{x-a}{b}+\frac{x-b}{a}=\frac{b}{x-a}+\frac{a}{x-b}$ and $x_{1}-x_{2}-x_{3}=c$,
then $a, b, c$ are in
(A) A.P
(B) $\mathrm{G} . \mathrm{P}_{4}$
(C) H.P.
(D) None of these

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Key Concepts

Vieta’s Formulas and Symmetric Functions of Roots

Vieta’s formulas relate the sums and products of the roots of a polynomial to its coefficients. Working with symmetric functions of the roots provides a method to form connections between the given parameters and the structure of the equation. This approach is particularly useful when the problem defines additional conditions or relationships among the roots.

Relations in Sequences and Progressions

This concept encompasses the definitions and properties of different types of progressions, including arithmetic progression (AP), geometric progression (GP), and harmonic progression (HP). Understanding these relationships is essential when establishing whether a set of numbers, often defined through algebraic expressions, conforms to one of these sequential patterns.

Transformation of Rational Equations to Polynomial Equations

This concept involves converting equations that contain rational expressions into polynomial equations by clearing denominators. Such transformations allow one to leverage the well-developed theory of polynomials to study properties of the equation, such as its roots, and to apply techniques like factorization and the use of Vieta’s formulas.

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