Question

If the lines x + 2ay + a = 0, x + 3by + b = 0,x + 4cy + c = 0 are concurrent, then a,b,c are in 1) A.P 2) G.P 3) H.P 4) A.G.P​

Answer 1

Correct option is

C

Harmonic progression

For given lines to be concurrent.

∣

∣

∣

∣

∣

∣

∣

∣

​

 

x  

1

​

 

x  

2

​

 

x  

3

​

 

​

 

y  

1

​

 

y  

2

​

 

y  

3

​

 

​

 

z  

1

​

 

z  

2

​

 

z  

3

​

 

​

 

∣

∣

∣

∣

∣

∣

∣

∣

​

=0

∣

∣

∣

∣

∣

∣

∣

∣

​

 

1

1

1

​

 

2a

3b

4c

​

 

a

b

c

​

 

∣

∣

∣

∣

∣

∣

∣

∣

​

=0

⇒1(3bc−4bc)−1(2ac−4ac)+1(2ab−3ab)=0

⇒−bc+2ac−ab

⇒2ac=bc+ab

⇒  

b

2

​

=  

a

1

​

+  

c

1

​

⇒a,b,c∈ H.P