Question

If the obtuse angle bisector and angular bisector contains origin is same for the pair of lines $2x+3y-\lambda =0$ and $3x+2y-(\lambda +2)=0$ then $\lambda$ belongs to

• A. $(-2,0)$
• B. $(-\infty ,-2)\cup (0,\infty )$
• C. $(-\infty ,-2]\cup [0,\infty )$
• D. None of the above

Answer 1

The correct option is B $(-\infty ,-2)\cup (0,\infty )$

$a_{1}x+b_{1}y+c_1=2x+3y-\lambda =0$
$a_{2}x+b_{2}y+c_2=3x+2y-(\lambda +2)=0$
here $a_1{a}_2+b_1{b}_2>0$
$\Rightarrow$obtuse angle bisector is $\frac{L_1}{\sqrt{a_1^2+b_1^2}}=+\frac{L_2}{\sqrt{a_2^2+b_2^2}}$
Now at origin
$L_1{L}_2=(-\lambda )(-(\lambda +2\left)\right)>0$
$\Rightarrow \lambda (\lambda +2)>0$
$\Rightarrow \lambda \in (-\infty ,-2)\cup (0,\infty )$