Determining Tangent Slope w/Point Not On Curve

  • Thread starter Millacol88
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How do you determine the equation of all possible tangents to a curve (say, a parabola) that pass through a given point that is not on said curve? This is more of a conceptual question, and it's not homework, so I thought it fit in this forum. I think there might be a question like this on the test tomorrow. :wink:
 

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You write down the line equation as l(x, t) = (x, f(x)) + t(1, f'(x)) and solve a system of equations l(x, t) = p, where p is the point you want to pass. You then have all values of x that have a tangent you want.
If the curve is given implicitly instead, as F(x, y) = 0, you can find a normal n(x, y) = (dF(x, y)/dx, dF(x, y)/dy). Then you write an implicit line equation <n(x, y), p> = <n(x, y), (x, y)> and get another implicit curve. Lastly, you need to find where this curve intersects with the given curve.
I'm not sure this is the easiest way though.
 

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