Question

Show that the area of the triangle formed by the lines `y=m_1x+c_1,y=m_2x+c_2`and `x=0`is `((c_1-c_2)^2)/(2|m_1-m_2|)`

Answer 1

Let the sides of AC, AB and BC of `triangle ABC` be represented by all the equations:
`m_(1)x-y+c_(1)=0........(i)`
`m_(2)x-y+c_(2)=0........(ii)`
`x=0........(iii)`
On solving (i) and (ii) by cross multiplication, we have
`(x)/((-c_(2)+c_(1)))=(y)/((m_(2)c_(1)-m_(1)c_(2)))=(1)/((-m_(1)+m_(2)))`
`Rightarrow x=((c_(1)-c_(2)))/((m_(2)-m_(1)))and y=((m_(2)c_(1)-m_(1)c_(2)))/((m_(2)-m_(1)))`
Thus, the lines AC and AB intersect at A`((c_(1)-c_(2))/(m_(2)-m_(1)),(m_(2)c_(1)-m_(1)c_(2))/(m_(2)-m_(1)))`
On solving (ii) and (iii), we get `B(0,c_(2))`
On solving (i) and (iii), we get `C(0,c_(1))`
`therefore "area of "triangle ABC "is given by "`
`triangle=(1)/(2).|x_(1)(y_(2)-y_(3))+x_(2)(y_(3)-y_(1))+x_(3)(y_(1)-y_(2))|`
`=(1)/(2).|((c_(1)-c_(2)))/((m_(2)-m_(1))).(c_(2)-c_(1))+0+0|=(1)/(2).((c_(1)-c_(2))^(2))/(|m_(1)-m_(2)|)`
Hence the area of the triangle formed by the given line is `(1)/(2).((c_(1)-c_(2))^(2))/(|m_(1)-m_(2)|)`