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 $\mathrm{AD}$ is:

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

Answer 1

The correct option is C

$x-2y=0$

Explanation for the correct option.

Step 1: Find the slope of line

Given that $\mathrm{D}$ is the mid-point of $\mathrm{BC}$:
$$\begin{gathered} \mathrm{D}=\left(\frac{0+2}{2},\frac{0+2}{2}\right) \\ \mathrm{D}=(1,1) \end{gathered}$$

The slope of line joining $\left({\mathrm{x}}_1,{\mathrm{y}}_1\right),\left({\mathrm{x}}_2,{\mathrm{y}}_2\right)$ is given by $\mathrm{m}=\frac{{\mathrm{y}}_2-{\mathrm{y}}_1}{{\mathrm{x}}_2-{\mathrm{x}}_1}$
Let the slope of $\mathrm{AD}$ be ${\mathrm{m}}_1$
$\begin{array}{rcl}\mathrm{m}& =& \frac{1-5}{1+1}\\ & =& -2\end{array}$
Let the slope of the required line be ${\mathrm{m}}_2$.
The required line is perpendicular to $\mathrm{AD}$.

Therefore, ${\mathrm{m}}_1{\mathrm{m}}_2=-1$
$\begin{array}{rcl}(-2){\mathrm{m}}_2& =& -1\\ {\mathrm{m}}_2& =& \frac{1}{2}\end{array}$

Step 2: Find the equation of the line
This line passes through $B$ is $\mathrm{y}-{\mathrm{y}}_1=\mathrm{m}\left(\mathrm{x}-{\mathrm{x}}_1\right)$.
$\begin{array}{rcl}\mathrm{y}-0& =& \frac{1}{2}(\mathrm{x}-0)\\ \mathrm{y}& =& \frac{{\displaystyle 1}}{{\displaystyle 2}}\mathrm{x}\\ \mathrm{x}-2\mathrm{y}& =& 0\end{array}$
Therefore, the required line is $x-2y=0$.
Hence, option C is correct.