First slide

Equation of line in Straight lines

Question

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

Moderate

A

A.P.
B

G.P.
C

H.P.
D

none of these

Solution

Since the given lines are concurrent
$|\begin{array}{ccc} 1 & 2 a & a \\ 1 & 3 b & b \\ 1 & 4 c & c \end{array}| = 0 \Rightarrow - b c + 2 a c - a b = 0$
$\Rightarrow b = \frac{2 a c}{a + c} \Rightarrow a, b, c$ are in H.P.

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