Question

Find the equation of a line which passes through the mid point of $(3,1)$ and $(1,5)$. The slope of this line is perpendicular to another line having a slope of $\frac{-1}{3}.$

• A. $y-3=3(x-2)$
• B. $y-2=3(x-3)$
• C. $y-3=\frac{-1}{3}(x-2)$
• D. $y-3=\frac{1}{3}(x-2)$

Answer 1

The correct option is A $y-3=3(x-2)$

We have to obtain the point through which the line is passing and the slope of this line.
Step 1: Find the point
The line passes through mid point of $(3,1)$ and $(1,5).$
The midpoint of the points $(x_1,y_1)$ and $(x_2,y_2)$ is given by
$(\frac{x_1+y_1}{2},\frac{y_1+y_2}{2})$

$\therefore$ The midpoint of (3, 1) and (1, 5)
$=(\frac{3+1}{2},\frac{1+5}{2})$
$=(\frac{4}{2},\frac{6}{2})$
$=(2,3)$
Hence, the line passes through $(2,3).$

Step 2: Find the slope of the line
The line is perpendicular to another line of slope $\frac{-1}{3}.$
Let the slope of the line is $m_1$ and slope of the perpendicular line is $m_2$
Given, $m_2=\frac{-1}{3}$

Concept: If two line are perpendicular to each other, then product of the slope of both line is $-1.$
$m_1\times m_2=-1$
$m_1\times \frac{-1}{3}=-1$
$m_1=\frac{-3}{-1}$
$m_1=3$

Step 3: The point slope form of a line is given as:

$y-y_1=m(x-x_1)$ where 'm' is the slope and $(x_1,y_1)$ is the point from which the line is passing.

Slope $=3$ and point $=(2,3)$

Equation of line: $y-3=3(x-2)$

Hence, option (a) is the correct choice.