Question

Consider two lines, $L_1: 2 x+y+4=0$ and $L_2: 2 x+4 y+5=0$, then

• A. Bisector of angle containing origin is $2x - 2y + 3 = 0.$
• B. Bisector of acute angle is $6x + 6y + 13 = 0$.
• C. Angle containing origin is acute angle.
• D. Angle containing origin is obtuse.

Answer 1

Answer: (A), (B), (D)

$ C _1, C _2$ are both positive
(1) Bisector of angle containing origin
$\frac{2 x+y+4}{\sqrt{5}}=\frac{2 x+4 y+5}{2 \sqrt{5}}$
$4x + 2y + 8 = 2x + 4y + 5$
$2x - 2y + 3 = 0$
(2) Since $a_1 a_2+b_1 b_2>0 \Rightarrow$ origin occurs in obtuse angle. bisector of acute angle
$\frac{2 x+y+4}{\sqrt{5}}=\frac{-(2 x+4 y+5)}{2 \sqrt{5}}$
$2x + y + 4 = -2x - 4x - 5$
$4x + 5y + 8 = -2x - 4 - 5$
$6x + 6y + 13 = 0$