Question

If the coordinates of points $A,B,C$ be $(-1,5),(0,0)$ and $(2,2)$ respectively and $D$ be the middle point of $BC$, then the equation of the perpendicular drawn from $B$ to the line $AD$ is:

• A. $2x+y=0$
• B. $x+2y=0$
• C. $x-2y=0$
• D. $2x-y=0$

Answer 1

The correct option is C $x-2y=0$

Given that $D$ is the mid point of $BC$

$\Rightarrow D=(\frac{0+2}{2},\frac{0+2}{2})$

$\Rightarrow D=(1,1)$

Slope of line joining $(x_1,y_1),(x_2,y_2)$ is given by $m=\frac{y_2-y_1}{x_2-x_1}$

Let Slope of $AD$ be $m_1$

$\Rightarrow m=\frac{1-5}{1+1}=\frac{-4}{2}=-2$

Let the slope of required line be $m_2$

Required line is perpendicular to $AD$ . Therefore, $m_1{m}_2=-1$

$\Rightarrow (-2)m_2=-1$

$\Rightarrow m_2=\frac{1}{2}$

This line passes through $B$

$\Rightarrow y-y_1=m(x-x_1)$

$\Rightarrow y-0=\frac{1}{2}(x-0)$

$\Rightarrow y=\frac{1}{2}x$

$\Rightarrow x-2y=0$

Hence the required line is $\Rightarrow x-2y=0$