Question

If the lines ax + 12y + 1 = 0, bx + 13y + 1 = 0 and cx + 14y + 1 = 0 are concurrent, then a, b, c are in
(a) H.P.
(b) G.P.
(c) A.P.
(d) none of these

Answer 1

(c) A.P.

The given lines are

ax + 12y + 1 = 0 ... (1)

bx + 13y + 1 = 0 ... (2)

cx + 14y + 1 = 0 ... (3)

It is given that (1), (2) and (3) are concurrent.

$$\begin{gathered} \left|\begin{array}{ccc}a& 12& 1\\ b& 13& 1\\ c& 14& 1\end{array}\right|=0 \\ \Rightarrow a\left(13-14\right)-12\left(b-c\right)+14b-13c=0 \\ \Rightarrow -a-12b+12c+14b-13c=0 \\ \Rightarrow -a+2b-c=0 \\ \Rightarrow 2b=a+c \end{gathered}$$

Hence, a, b and c are in AP.