Question

The line $l$ goes through the points $(1,2)$ and $(5,10)$. Which of the following could be the equation of a line perpendicular to $l$?

• A. $4x+2y=6$
• B. $2x-y=0$
• C. $x+y=3$
• D. $x+2y=5$
• E. $x-2y=5$

Answer 1

Answer: (D)

$x+2y=5$

The slope of the line passing through the points $(x_1,y_1)$ and $(x_2,y_2)$ is:

$m=\frac{y_2-y_1}{x_2-x_1}$

Therefore, the slope of the line passing through the points $(1,2)$ and $(5,10)$ is:

$m=\frac{10-2}{5-1}=\frac{8}{4}=2$

Here, the line is perpendicular to $l$, therefore, the slope of $l$ is the negative reciprocal of $2$, that is $-\frac{1}{2}$.

Equation of line with slope $m$ and a point $(x_1,y_1)$ is $y-y_1=m(x-x_1)$,

Therefore, equation of the line with slope $m=-\frac{1}{2}$ and point $(1,2)$ is given by:

$y-2=-\frac{1}{2}(x-1)\Rightarrow y-2=-\frac{1}{2}x+\frac{1}{2}\Rightarrow 2(y-2)=-x+1\Rightarrow 2y-4=-x+1\Rightarrow 2y+x=1+4\Rightarrow 2y+x=5$

Hence, the equation of the line is $x+2y=5$.