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

Prove that, area of the triangle formed by three straight lines, $a_{1} x + b_{1} y + c_{1} = 0, a_{2} x + b_{2} y + c_{2} = 0$ and $a_{3} x + b_{3} y + c_{3} = 0$ is $\frac{1}{2} \frac{{|\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}|}^{2}}{(a_{1} b_{2} - b_{1} a_{2}) (a_{2} b_{3} - b_{2} a_{3}) (a_{3} b_{1} - b_{3} a_{1})}$ Sq. unit.

Accepted Answer

Given, equation of three straight lines,

$a_{1} x + b_{1} y + c_{1} = 0 . . . (i)$

$a_{2} x + b_{2} y + c_{2} = 0 . . . (i i)$

$a_{3} x + b_{3} y + c_{3} = 0 . . . (i i i)$

Let $(α_{1}, β_{1}), (α_{2}, β_{2})$ and $(α_{3}, β_{3})$ be the vertices of triangle formed by those lines.

Using cross-multiplication method to find the value of $x$ and $y$ from equation $(i)$ and $(i i)$ .

$\frac{x}{|\begin{matrix} b_{1} & c_{1} \\ b_{2} & c_{2} \end{matrix}|} = \frac{y}{|\begin{matrix} c_{1} & a_{1} \\ c_{2} & a_{2} \end{matrix}|} = \frac{1}{|\begin{matrix} a_{1} & b_{1} \\ a_{2} & b_{2} \end{matrix}|}$

$\Rightarrow x = \frac{|\begin{matrix} b_{1} & c_{1} \\ b_{2} & c_{2} \end{matrix}|}{|\begin{matrix} a_{1} & b_{1} \\ a_{2} & b_{2} \end{matrix}|}$

and $y = \frac{|\begin{matrix} c_{1} & a_{1} \\ c_{2} & a_{2} \end{matrix}|}{|\begin{matrix} a_{1} & b_{1} \\ a_{2} & b_{2} \end{matrix}|}$

So $(α_{1}, β_{1})$ , coordinates of points of intersection of equations $(i)$ and $(i i)$ will be,

$α_{1} = \frac{|\begin{matrix} b_{1} & c_{1} \\ b_{2} & c_{2} \end{matrix}|}{|\begin{matrix} a_{1} & b_{1} \\ a_{1} & b_{2} \end{matrix}|}, β_{1} = \frac{|\begin{matrix} c_{1} & a_{1} \\ c_{2} & a_{2} \end{matrix}|}{|\begin{matrix} a_{1} & b_{1} \\ a_{1} & b_{2} \end{matrix}|}$

Similarly, If $(α_{2}, β_{2})$ are coordinates of the intersection of equations $(i i)$ and $(i i i)$ ,

then $(α_{2}, β_{2}) \equiv (\frac{|\begin{matrix} b_{2} & c_{2} \\ b_{3} & c_{3} \end{matrix}|}{|\begin{matrix} a_{2} & b_{2} \\ a_{3} & b_{3} \end{matrix}|}, \frac{|\begin{matrix} c_{2} & a_{2} \\ c_{3} & a_{3} \end{matrix}|}{|\begin{matrix} a_{2} & b_{2} \\ a_{3} & b_{3} \end{matrix}|})$

and if $(α_{3}, β_{3})$ are coordinates of the intersection point of equations $(i i i)$ and $(i)$ ,

then $(α_{3}, β_{3}) \equiv (\frac{|\begin{matrix} b_{3} & c_{3} \\ b_{1} & c_{1} \end{matrix}|}{|\begin{matrix} a_{3} & b_{3} \\ a_{1} & b_{1} \end{matrix}|}, \frac{|\begin{matrix} c_{2} & a_{2} \\ c_{3} & a_{3} \end{matrix}|}{|\begin{matrix} a_{3} & b_{3} \\ a_{1} & b_{1} \end{matrix}|})$

Area of the triangle, having vertices $(α_{1}, β_{1}), (α_{2}, β_{2})$ and $(α_{3}, β_{3})$

$= \frac{1}{2} |\begin{matrix} α_{1} & β_{1} & 1 \\ α_{2} & β_{2} & 1 \\ α_{3} & β_{3} & 1 \end{matrix}|$

$= \frac{1}{2} |\begin{matrix} \frac{|\begin{matrix} b_{1} & c_{1} \\ b_{2} & c_{2} \end{matrix}|}{|\begin{matrix} a_{1} & b_{1} \\ a_{2} & b_{2} \end{matrix}|} & \frac{|\begin{matrix} c_{1} & a_{1} \\ c_{2} & a_{2} \end{matrix}|}{|\begin{matrix} a_{1} & b_{1} \\ a_{2} & b_{2} \end{matrix}|} & 1 \\ \frac{|\begin{matrix} b_{1} & c_{2} \\ b_{2} & c_{3} \end{matrix}|}{|\begin{matrix} a_{2} & b_{2} \\ a_{3} & b_{3} \end{matrix}|} & \frac{|\begin{matrix} c_{2} & a_{2} \\ c_{3} & a_{3} \end{matrix}|}{|\begin{matrix} a_{2} & b_{2} \\ a_{3} & b_{3} \end{matrix}|} & 1 \\ \frac{|\begin{matrix} b_{3} & c_{3} \\ b_{1} & c_{1} \end{matrix}|}{|\begin{matrix} a_{3} & b_{3} \\ a_{1} & b_{1} \end{matrix}|} & \frac{|\begin{matrix} c_{3} & a_{3} \\ c_{1} & a_{1} \end{matrix}|}{|\begin{matrix} a_{3} & b_{3} \\ a_{1} & b_{1} \end{matrix}|} & 1 \end{matrix}|$

$= \frac{1}{2} (\frac{1}{|\begin{matrix} a_{1} & b_{1} \\ a_{2} & b_{2} \end{matrix}|} .) (\frac{1}{|\begin{matrix} a_{2} & b_{2} \\ a_{3} & b_{3} \end{matrix}|}) . (\frac{1}{|\begin{matrix} a_{3} & b_{3} \\ a_{1} & b_{1} \end{matrix}|})$

$|\begin{matrix} |\begin{matrix} b_{1} & c_{2} \\ b_{2} & c_{1} \end{matrix}| & |\begin{matrix} c_{1} & a_{1} \\ c_{2} & a_{2} \end{matrix}| & |\begin{matrix} a_{1} & b_{1} \\ a_{2} & b_{2} \end{matrix}| \\ |\begin{matrix} b_{2} & c_{2} \\ b_{3} & c_{3} \end{matrix}| & |\begin{matrix} c_{2} & a_{2} \\ c_{3} & a_{3} \end{matrix}| & |\begin{matrix} a_{2} & b_{2} \\ a_{3} & b_{3} \end{matrix}| \\ |\begin{matrix} b_{3} & c_{3} \\ b_{1} & c_{1} \end{matrix}| & |\begin{matrix} c_{3} & a_{3} \\ c_{1} & a_{1} \end{matrix}| & |\begin{matrix} a_{3} & b_{3} \\ a_{1} & b_{1} \end{matrix}| \end{matrix}|$

On simplifying we get,

$= \frac{1}{2} \frac{{|\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}|}^{2}}{(a_{1} b_{2} - b_{1} a_{2}) (a_{2} b_{3} - b_{2} a_{3}) (a_{3} b_{1} - b_{3} a_{1})}$ sq. unit.

S L Loney Solutions for Chapter: The Straight Line (Continued), Exercise 1: EXAMPLES X

S L Loney Mathematics Solutions for Exercise - S L Loney Solutions for Chapter: The Straight Line (Continued), Exercise 1: EXAMPLES X

Questions from S L Loney Solutions for Chapter: The Straight Line (Continued), Exercise 1: EXAMPLES X with Hints & Solutions

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