# Tangent and parallel lines

• March 12th 2008, 08:20 AM
hasanbalkan
Tangent and parallel lines
Find all lines that are tangent to the curve y = x^3 and are also parallel to the line 3x - y = 5

Any help will be appreciated!
• March 12th 2008, 08:42 AM
wingless
Quote:

Originally Posted by hasanbalkan
Find all lines that are tangent to the curve y = x^3 and are also parallel to the line 3x - y = 5

Any help will be appreciated!

Firstly,
$y=x^3$
$y'=3x^2$

The lines must be parallel to 3x-y = 5, $y=3x-5$. So their slope must be $m=3$.

Let our line be $y=m\cdot x + C$, m is the slope and C is a constant. As $m=3$ it'll become $y=3x + C$

At the tangent point, the line must satisfy $m=3=3x^2$.
$3x^2 = 3$
$x^2=1$
$x= \pm 1$

So the line passes the points (-1,f(-1)) and (1,f(1)).
(-1,-1), (1,1)

The line is $y=3x + C$
And it makes $C=2$ for (-1,1)
and $C=-2$ for (-1,1)

The lines are then
$y=3x + 2$ and $y=3x - 2$