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Geometric Mean

If the line ...

Question

If the line $x + 2 a y + a = 0, x + 3 b y + b = 0, x + 4 c y + c = 0$ are concurrent,then a,b,c are in.

Arithmetic progression

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Geometric progression

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Harmonic progression

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Arithmetico-Geometric progression

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Solution

The correct option is C Harmonic progression
For given lines to be concurrent.

$∣ ∣ ∣ ∣ \begin{matrix} x_{1} & y_{1} & z_{1} x_{2} & y_{2} & z_{2} x_{3} & y_{3} & z_{3} \end{matrix} ∣ ∣ ∣ ∣ = 0$

$∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 a & a 1 & 3 b & b 1 & 4 c & c \end{matrix} ∣ ∣ ∣ ∣ = 0$

$\Rightarrow 1 (3 b c - 4 b c) - 1 (2 a c - 4 a c) + 1 (2 a b - 3 a b) = 0$

$\Rightarrow - b c + 2 a c - a b$

$\Rightarrow 2 a c = b c + a b$

$\frac{2}{b} = \frac{1}{a} + \frac{1}{c} \Rightarrow a, b, c$ are in $H P$

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