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  • If the straight lines, <img alt="a x+a m y+1=0, b x+(m+1) b y+1=0 \\" src="https://entrancecorner.oncodecogs.com/gif.latex?a%20x&plus;a%20m%20y&plus;1%3D0%2C%20b%20x&plus

If the straight lines, a x+a m y+1=0, b x+(m+1) b y+1=0 \\
and c x+(m+2) c y+1=0, m \neq 0  are concurrent, then a, b, c are in

 

 

Option: 1

AP only form=1


Option: 2

 AP for all m


Option: 3

 GP for all m


Option: 4


 HP for all m


Answers (1)

best_answer

\begin{gathered} D=\left|\begin{array}{ccc} a & a m & 1 \\ b & b(m+1) & 1 \\ c & c(m+2) & 1 \end{array}\right|=0 ;\left|\begin{array}{ccc} a & 0 & 1 \\ b & b & 1 \\ c & 2 c & 1 \end{array}\right|=0 \\ a(b-2 c)+(2 b c-b c)=0 \\ \Rightarrow b-2 a c+b c=0 \\ \Rightarrow \quad b(a+c)=2 a c \\ \Rightarrow \quad a, b, c \text { are in HP. }\\ \end{gathered} \\ \Rightarrow a, b, c \text{are in } HP.

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