A line passing through origin and is perpendicular to two given l… - Vidyadip

MCQ

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

A
$1 : 2$
B
$2 : 1$
✓
$4 : 3$
D
$3 : 4$

✓

Answer

Correct option: C.

$4 : 3$

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

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

Now say origin divide the line $x - 2y = 0$ 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 $x = 2y$, in the ratio $4 : 3$.

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