perpendicular lines

Quote:

Originally Posted by muddywaters

the question with the answers:
Attachment 24153
Attachment 24154
Attachment 24155

what i don't understand is this part:
Attachment 24156
how did that happen?

1. You know from equation 2:

$\displaystyle{y-y_0=-\frac1m(x-x_0)~\implies~(y-y_0)^2=\frac1{m^2}(x-x_0)^2}$

2. $\displaystyle{|\overline{PQ}|=\sqrt{(x-x_0)^2+(y-y_0)^2}}$

Replace $\displaystyle{(y-y_0)^2}$ by the term of #1:

$\displaystyle{|\overline{PQ}|=\sqrt{(x-x_0)^2+\frac1{m^2}(x-x_0)^2}}$

$\displaystyle{|\overline{PQ}|=\sqrt{\frac{m^2}{m^2 } (x-x_0)^2+\frac1{m^2}(x-x_0)^2}}$

Factor out $\displaystyle{(x-x_0)^2}$

$\displaystyle{|\overline{PQ}|=\sqrt{\frac{m^2-1}{m^2} (x-x_0)^2}}$