Paragraph: If Δ(a, b, c, α)=0 for every α > 0 , then which one of the following is correct?

Correct answer: B. a, b, c are in GP | array ccc a & b & a α+b b & c & b α+c a α+b & b α+c & 0 array |=0R_3 arrow R_3-α R_1-R_2| array ccc a & b & a α+b b & c & b α+c 0 & 0 & -a α^2-b α-b α-c array |=0 Expanding along R_3 , -(a α^2+2 b α+c)(b^2-a c)=0α ≠ 0b^2-a c=0 b^2=a ca, b, c are in GP

NDA 2023

Exam: NDA
Year: 2023
Subject: Mathematics
Chapter: Determinants

NDA 2023 · Mathematics · Determinants

Paragraph: If Δ(a, b, c, α)=0 for every α > 0 , then which one of...

Paragraph: If

\Delta(a, b, c, \alpha)=0

for every

\alpha > 0

, then which one of the following is correct?

Options

A $a, b, c$ are in AP
B $a, b, c$ are in GP
C $a, 2 b, c$ are in AP
D $a, 2 b, c$ are in GP

Correct answer

B. $a, b, c$ are in GP

Step-by-step solution

\left|

\begin{array}{ccc}a & b & a \alpha+b \\ b & c & b \alpha+c \\ a \alpha+b & b \alpha+c & 0\end{array}

\right|=0R_3 \rightarrow R_3-\alpha R_1-R_2\left|

\begin{array}{ccc}a & b & a \alpha+b \\ b & c & b \alpha+c \\ 0 & 0 & -a \alpha^2-b \alpha-b \alpha-c\end{array}

\right|=0

Expanding along

R_3

-\left(a \alpha^2+2 b \alpha+c\right)\left(b^2-a c\right)=0\alpha \neq 0b^2-a c=0 \Rightarrow b^2=a ca, b, c

are in GP

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