Column - I Column - II
A) If the lines $x + 2 a y + a = 0, x + 3 b y + b = 0$ $, x + 4 c y + c = 0$ where a, b, c $\in R$ are concurrent, then a,b,c are in p) A.P
B) The point with coordinates (2a, 3a), (3b, 2b), (c, c) where a, b, c $\in R$ are collinear, then a, b, c are in q) G.P
C) If lines $a x + 2 y + 1 = 0, b x + 3 y + 1 = 0$ and $c x + 4 y + 1 = 0$ where a, b, c $\in ℝ$ passes through the same point, then a, b, c are in r) H.P
D) Let a, b, c be distinct non-negative real numbers. If the lines $a x + a y + c = 0, x + 1 = 0,$ $c x + c y + b = 0$ pass through the same point then a, b, c are in s) Neither A.P. nor G.P. nor H.P

Moderate

Solution

A) $|\begin{matrix} 1 & 2 a & a \\ 1 & 3 b & b \\ 1 & 4 c & c \end{matrix}| = 0 \Rightarrow \frac{2}{b} = \frac{1}{a} + \frac{1}{c}$
B) $|\begin{matrix} 2 a & 3 a & 1 \\ 3 b & 2 b & 1 \\ c & c & 1 \end{matrix}| = 0 \Rightarrow b c + a - 5 a b = 0$
C) $|\begin{matrix} a & 2 & 1 \\ b & 3 & 1 \\ c & 4 & 1 \end{matrix}| = 0 \Rightarrow a + c = 2 b$
D) $|\begin{matrix} a & a & 0 \\ c & c & b \\ 1 & 0 & 1 \end{matrix}| = 0 \Rightarrow c^{2} = a b$

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