Name: Angle Between Two Lines
Uploaded: 2019-07-19
Duration: 6 min 13 s
Description: If two lines are perpendicular to each other, their direction vectors are also perpendicular. This means the scalar product of the direction vectors is equal...

Question

Find the area of the triangle formed by the lines $y = m_{1} x + c_{1}, y = m_{2} x + c_{2} and x = 0$ .

Accepted Answer

Given that the following lines represents the sides of the $△ A B C$ .

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

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

$x = 0 . . . (3)$

In triangle $A B C$ , let equations $(1), (2)$ and $(3)$ represent the sides $A B, B C$ and $C A,$ respectively.

Solving $(1)$ and $(2)$ using the cross multiplication method, we have

$\frac{x}{c_{1} - c_{2}} = \frac{y}{m_{2} c_{1} - m_{1} c_{2}} = \frac{1}{m_{2} - m_{1}}$

$x = \frac{c_{2} - c_{1}}{m_{1} - m_{2}}, y = \frac{m_{1} c_{2} - m_{2} c_{1}}{m_{1} - m_{2}}$

Thus, $A B$ and $B C$ intersect at $B$ $(\frac{c_{2} - c_{1}}{m_{1} - m_{2}}, \frac{m_{1} c_{2} - m_{2} c_{1}}{m_{1} - m_{2}}) .$

Solving $(1)$ and $(3) :$
$x = 0, y = c_{1}$

Thus, $A B$ and $C A$ intersect at $A$ $(0, c_{1})$ .

Similarly, solving $(2)$ and $(3) :$
$x = 0, y = c_{2}$

Thus, $B C$ and $C A$ intersect at $C$ $(0, c_{2})$ .

$∴ Area of triangle A B C = \frac{1}{2} |\begin{matrix} 0 & c_{1} & 1 \\ 0 & c_{2} & 1 \\ \frac{c_{2} - c_{1}}{m_{1} - m_{2}} & \frac{m_{1} c_{2} - m_{2} c_{1}}{m_{1} - m_{2}} & 1 \end{matrix}|$

$= \frac{1}{2} [0 - c_{1} \frac{(c_{2} - c_{1})}{(m_{1} - m_{2})} + 1 (0 - c_{2} \frac{(c_{2} - c_{1})}{(m_{1} - m_{2})})]$

$= \frac{1}{2} (\frac{{c_{1}}^{2} - {c_{2}}^{2}}{m_{1} - m_{2}})$

Find the area of the triangle formed by the lines $y = m_{1} x + c_{1}, y = m_{2} x + c_{2} and x = 0$ .

Important Questions on The Straight Lines

Learn and Prepare for any exam you want !

Find the area of the triangle formed by the lines y=m1x+c1,y=m2x+c2 and x=0.

Important Questions on The Straight Lines

Learn and Prepare for any exam you want !

Find the area of the triangle formed by the lines $y = m_{1} x + c_{1}, y = m_{2} x + c_{2} and x = 0$ .