Question

If the lines $2x-py+1=0,3x-qy+1=0$ and $4x-ry+1=0$ are concurrent, then $p,q,r$ are in

• A. $A.P.$
• B. $G.P.$
• C. $H.P.$
• D. $A.G.P.$

Answer 1

The correct option is A $A.P.$

Let $(x_1,y_1)$ be the point of concurrence

$2x_1-p{y}_1+1=0\Rightarrow p{y}_1=2x_1+13x_1-q{y}_1+1=0\Rightarrow q{y}_1=3x_1+14x_1-r{y}_1+1=0\Rightarrow r{y}_1=4x_1+1$

Now,
$p{y}_1+r{y}_1=(2x_1+1)+(4x_1+1)\Rightarrow p{y}_1+r{y}_1=6x_1+2\Rightarrow (p+r)y_1=2(3x_1+1)=2q{y}_1\Rightarrow p+r=2q\therefore p,q,r\to A.P$