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Thread: Tangent and parallel lines

  1. #1
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    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!
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  2. #2
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    Quote Originally Posted by hasanbalkan View Post
    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,
    $\displaystyle y=x^3$
    $\displaystyle y'=3x^2$

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

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

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

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

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

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

    bu arada, türk müsün? =)
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