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Homework Help: Finding tangent lines to a parabola.

  1. Oct 9, 2006 #1
    Using a graph to help you, find the equations of all lines through the origin tangent to the parabola

    Sketch the lines on the graph.

    I'm not sure how to start this off, I know the the first derivative would give me the slope of the tangent line at any point. If I find that:

    Now, I know that my line has the slope [tex]2x-2[/tex] and it goes through the origin. So, I thought of plugging that into the slope-intercept form for a line, but it doesn't really work, I don't think:

    The [tex]x[/tex]'s go out, and I'm left with an equation of:

    When I graphed this parabola and the original one, I noticed that the graphs intersected at the points that the tangent lines would go through. So, to find those points, I set the equations equal to each other:

    Since I just got the x-values, I plugged them in to the original function to get the corresponding y-values:

    Now, I can find the equations of the two tangent lines by using the points I found:
    First tangent line:
    (0, 0), (2, 4)

    Second tangent line:
    (0, 0), (-2, 12)

    Is this correct? It seems like there would be an easier way to go about doing this... Does anyone know?... It seems like I made things work in some very weird, rough way.
    Last edited: Oct 9, 2006
  2. jcsd
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