Question

Two perpendicular lines have opposite y-intercepts. The equation of one of these lines is

Answer 1

Answer:  The required x-co-ordinate of the point of intersection of two lines is $-\dfrac{2bm}{m^2+1}.$

Step-by-step explanation:  Given that two perpendicular lines have opposite y-intercept and the equation of one of the lines is

$y=mx+b.$

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

$ms=-1~~~\Rightarrow s=-\dfrac{1}{m}$

and c = -b.

That is, the equation of the other line is

$y=sx+c\\\\\Rightarrow y=-\dfrac{1}{m}-b.$

Comparing the equations of both the lines, we get

$mx+b=-\dfrac{1}{m}x-b\\\\\\\Rightarrow mx+\dfrac{1}{m}x=-2b\\\\\\\Rightarrow \dfrac{m^2+1}{m}x=-2b\\\\\\\Rightarrow x=-\dfrac{2bm}{m^2+1}.$

Thus, the required x-co-ordinate of the point of intersection of two lines is $-\dfrac{2bm}{m^2+1}.$