Question

A line passing through origin and is perpendicular to two given lines $2 x + y + 6 = 0$ and $4 x + 2 y - 9 = 0$, then the ratio in which the origin divides this line is.

• A. 1 : 2
• B. 2 : 1
• C. 4 : 3
• D. 3 : 4

Answer 1

Answer: (C)

4 : 3

Answer 2

Equation of line Perpendicular to $2 x + y + 6 = 0$ passes through (0, 0) is $x - 2 y = 0$

Now point of intersection of $x - 2 y = 0$ and $2 x + y + 6 = 0$is $\left( \frac { - 12 } { 5 } , \frac { - 6 } { 5 } \right)$ and point of intersection of $x - 2 y = 0$ and $4 x + 2 y - 9 = 0$ is $\left( \frac { 9 } { 5 } , \frac { 9 } { 10 } \right)$.

Now say origin divide the line $x - 2 y = 0$ in the ratio $\lambda : 1$

∴ $x = \frac { \frac { 9 } { 5 } \lambda - \frac { 12 } { 5 } } { \lambda + 1 } = 0 \Rightarrow \frac { 9 } { 5 } \lambda = \frac { 12 } { 5 }$, $\therefore \lambda = \frac { 4 } { 3 }$

Thus origin divides the line $x = 2 y$, in the ratio 4 : 3.