Name: The Line Family
Uploaded: 2019-07-19
Duration: 4 min 33 s
Description: A family of lines is a set of lines that have something in common with each other. Learn more about the family of lines by watching this video.

Question

Find area of the triangle formed by the straight lines whose equations are

y = m_{1} x + c_{1}, y = m_{2} x + c_{2}

and

y = m_{3} x + c_{3}

Accepted Answer

Given equation of straight lines,

$y = m_{1} x + c_{1} . . . (1)$

$y = m_{2} x + c_{2} . . . (2)$

$y = m_{3} x + c_{3} . . . (3)$

Coordinates of vertex $C$ is the point of intersection of equation $(1)$ and $(2)$ : $(\frac{c_{2} - c_{1}}{m_{1} - m_{2}}, \frac{m_{1} c_{2} - m_{2} c_{1}}{m_{1} - m_{2}})$

Coordinates of vertex $A$ is the point of intersection of equation $(2)$ and $(3)$ : $(\frac{c_{3} - c_{2}}{m_{2} - m_{3}}, \frac{m_{2} c_{3} - m_{3} c_{2}}{m_{2} - m_{3}})$

Coordinates of vertex $B$ is the point of intersection of equation $(3)$ and $(1)$ : $(\frac{c_{1} - c_{3}}{m_{3} - m_{1}}, \frac{m_{3} c_{1} - m_{1} c_{3}}{m_{3} - m_{1}})$ .

Area of triangle having vertices coordinates

$(x_{1}, y_{1}), (x_{2}, y_{2})$ and $(x_{3}, y_{3})$ is given by

$\frac{1}{2} |\begin{matrix} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{matrix}|$

So, area of triangle, with vertices

$C (\frac{c_{2} - c_{1}}{m_{1} - m_{2}}, \frac{m_{1} c_{2} - m_{2} c_{1}}{m_{1} - m_{2}}), A (\frac{c_{3} - c_{2}}{m_{2} - m_{3}}, \frac{m_{2} c_{3} - m_{3} c_{2}}{m_{2} - m_{3}}), B (\frac{c_{1} - c_{3}}{m_{3} - m_{1}}, \frac{m_{3} c_{1} - m_{1} c_{3}}{m_{3} - m_{1}})$ is

$\frac{1}{2} |\begin{matrix} \frac{c_{2} - c_{1}}{m_{1} - m_{2}} & \frac{m_{1} c_{2} - m_{2} c_{1}}{m_{1} - m_{2}} & 1 \\ \frac{c_{3} - c_{2}}{m_{2} - m_{3}} & \frac{m_{2} c_{3} - m_{3} c_{2}}{m_{2} - m_{3}} & 1 \\ \frac{c_{1} - c_{3}}{m_{3} - m_{1}} & \frac{m_{3} c_{1} - m_{1} c_{3}}{m_{3} - m_{1}} & 1 \end{matrix}|$

$\Rightarrow \frac{1}{2} |[\frac{c_{2} - c_{1}}{m_{1} - m_{2}} (c_{1} - c_{2}) + \frac{c_{3} - c_{2}}{m_{2} - m_{3}} (c_{2} - c_{3}) + \frac{c_{1} - c_{3}}{m_{3} - m_{1}} (c_{3} - c_{1})]|$ sq.unit

$\frac{1}{2} |[\frac{{(c_{1} - c_{2})}^{2}}{m_{1} - m_{2}} + \frac{(c_{2} - c_{3})^{2}}{m_{2} - m_{3}} + \frac{(c_{3} - c_{1})^{2}}{m_{3} - m_{1}}]|$ sq. unit

Find area of the triangle formed by the straight lines whose equations are $y = m_{1} x + c_{1}, y = m_{2} x + c_{2}$ and $y = m_{3} x + c_{3}$

Important Questions on The Straight Line (Continued)

Learn and Prepare for any exam you want !

Find area of the triangle formed by the straight lines whose equations are y=m1x+c1, y=m2x+c2 and y=m3x+c3

Important Questions on The Straight Line (Continued)

Learn and Prepare for any exam you want !

Find area of the triangle formed by the straight lines whose equations are $y = m_{1} x + c_{1}, y = m_{2} x + c_{2}$ and $y = m_{3} x + c_{3}$