Q119 / 170 · Manipal (MET) 2016 · Mathematics

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

  • A
    $1: 2$
  • B
    $2: 1$
  • C
    $4: 2$
  • Correct answer
    D
    $4: 3$

Solution

Equation of line perpendicular to $2 x+y+6=0$ and passes through origin is $x-2 y=0$ Now, the point of intersection of $2 x+y+6=0$ and $x-2 y=0$ is $\left(-\frac{12}{5},-\frac{6}{5}\right)$ Similarly, the point of intersection of $x-2 y=0$ and $4 x+2 y-9=0$ is $\left(\frac{9}{5}, \frac{9}{10}\right)$. Let the origin divide the line $x-2 y=0$ in the ratio $\lambda: 1$ $\therefore \quad x=\frac{\frac{9}{5} \lambda-\frac{12}{5}}{\lambda+1}=0$ $\begin{array}{ll}\Rightarrow & \frac{9}{5} \lambda=\frac{12}{5} \\ \Rightarrow & \lambda=\frac{12}{9}=\frac{4}{3}\end{array}$

Hence, the correct answer is option: 'D'.

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