The system of equations a x+b y+(a α+b) z=0 b x+c y+(b α+c) z=0 (a α+b) x+(b α+c) y=0 has trivia

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The system of equations a x+b y+(a α+b) z=0 b x+c y+(b α+c)...

The system of equations $a x+b y+(a \alpha+b) z=0$ $b x+c y+(b \alpha+c) z=0$ $(a \alpha+b) x+(b \alpha+c) y=0$ has trivial solutions only. Then (a) $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in GP (b) $\alpha$ is a root of the equation $a x^{2}+b x+c=0$ (c) $\mathrm{x}-\alpha$ is a factor of $\mathrm{ax}^{2}+2 \mathrm{bx}+\mathrm{c}=0$ (d) $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{HP}$

The system of equations $a x+b y+(a \alpha+b) z=0$
$b x+c y+(b \alpha+c) z=0$
$(a \alpha+b) x+(b \alpha+c) y=0$ has trivial solutions only. Then
(a) $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in GP
(b) $\alpha$ is a root of the equation $a x^{2}+b x+c=0$
(c) $\mathrm{x}-\alpha$ is a factor of $\mathrm{ax}^{2}+2 \mathrm{bx}+\mathrm{c}=0$
(d) $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{HP}$

Key Concepts

Quadratic Equations

Quadratic equations, which are polynomials of degree two, are fundamental in algebra. They can be analyzed through various techniques such as factoring, completing the square, or using the quadratic formula. Understanding the roots of a quadratic provides insights into the behavior of related algebraic expressions and systems.

Factor Theorem

The factor theorem is a principle that connects zeros of polynomials with factors. It states that if a polynomial f(x) has a root at x = r, then (x - r) is a factor of f(x). This theorem is crucial for factoring polynomials, simplifying equations, and uncovering relationships between coefficients and roots.

Harmonic Progression (HP)

A harmonic progression is a sequence of numbers whose reciprocals form an arithmetic progression. This concept often appears in algebra when dealing with relationships among coefficients, and it helps in identifying specific proportional relationships between them.

Geometric Progression (GP)

A geometric progression is a sequence in which each term after the first is obtained by multiplying the previous term by a constant ratio. In problems involving coefficients or parameters, recognizing a GP can simplify analysis and help uncover relationships between variables or constants in algebraic contexts.

Determinant Condition for Uniqueness

For a homogeneous system, the existence of only the trivial solution is ensured when the determinant of the coefficient matrix is non-zero. This condition is pivotal in linear algebra as it guarantees that the system has a unique solution, meaning that the columns (or rows) of the matrix are linearly independent.

Homogeneous Systems of Linear Equations

A homogeneous system of linear equations is one where all the constant terms are zero. Such systems always have at least the trivial solution, where all unknowns equal zero, and the existence of non?trivial solutions depends on whether the coefficient matrix is singular (determinant zero) or non-singular (determinant non-zero). The uniqueness of the trivial solution indicates that the coefficient matrix is invertible.

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