# Thread: Tangent Lines, Parallel Lines, and Normal Lines

1. ## Tangent Lines, Parallel Lines, and Normal Lines

Find an equation of each line normal to the graph y=2x/(x-1) and parallel to the line 2x-y+1=0

2. Pretty sure it would just be y = 2x

It runs parallel to y=2x + 1 and would be perpendicular to the tangent lines at those points.

Are you supposed to find one line that will apply to both graphs?

3. Originally Posted by kinana18
Find an equation of each line normal to the graph y=2x/(x-1) and parallel to the line 2x-y+1=0
$\displaystyle y = \frac{2x}{x - 1} = 2 + \frac{2}{x-1} \Rightarrow \frac{dy}{dx} = \frac{-2}{(x-1)^2}$.

You want the normal to be parallel to $\displaystyle 2x - y + 1 = 0 \Rightarrow y = 2x + 1$.

To find the x-coordinates of the required normal solve $\displaystyle \frac{(x-1)^2}{2} = 2$: x = 0, 2.

Therefore the required normal is normal to the curve at (0, 0) and (2, 4). The equation of this normal is y = 2x.

Note: The normal is also the line of symmetry of the curve.