Answer: The required x-co-ordinate of the point of intersection of two lines is [tex]-\dfrac{2bm}{m^2+1}.[/tex]
Step-by-step explanation: Given that two perpendicular lines have opposite y-intercept and the equation of one of the lines is
[tex]y=mx+b.[/tex]
We are to express the x-coordinate of the intersection point of the lines in terms of m and b.
Let the slope and y-intercept of the other line be s and c respectively.
Since the product of the slopes of two perpendicular lines is -1 and -b is the opposite of b, so we have
[tex]ms=-1~~~\Rightarrow s=-\dfrac{1}{m}[/tex]
and c = -b.
That is, the equation of the other line is
[tex]y=sx+c\\\\\Rightarrow y=-\dfrac{1}{m}-b.[/tex]
Comparing the equations of both the lines, we get
[tex]mx+b=-\dfrac{1}{m}x-b\\\\\\\Rightarrow mx+\dfrac{1}{m}x=-2b\\\\\\\Rightarrow \dfrac{m^2+1}{m}x=-2b\\\\\\\Rightarrow x=-\dfrac{2bm}{m^2+1}.[/tex]
Thus, the required x-co-ordinate of the point of intersection of two lines is [tex]-\dfrac{2bm}{m^2+1}.[/tex]
