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Math Help - Help proving the formula for a tangent line of a parabola

  1. #1
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    Help proving the formula for a tangent line of a parabola

    We started learning about conic sections last week, and our assignment asks us to prove that, for the parabola

    y^2=4px

    the tangent line at point P(x1, y1) can be written as so:

    y1*y=2p(x+x1)

    I'm a little confused about how to proceed. I have that the slope will be

    dx/dy = y/2p

    but am unsure where to go from here.

    Thanks!
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  2. #2
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    Re: Help proving the formula for a tangent line of a parabola

    What you get from dy/dx is the slope of the tangent line to the parabola. Once you find the slope you will need to find the equation of the line of the form y = mx + b where m is the slope dy/dx. Your will find b using the coordinates of P in the equation of the line. You need to calculate the derivative correctly. Refer to your book on how to calculate the derivative.
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  3. #3
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    Re: Help proving the formula for a tangent line of a parabola

    From the given, you have  2y\frac{dy}{dx}=4p \Rightarrow \frac{dy}{dx}=\frac{2p}{y}
    Tangent line is given by:  y=mx+b
    Using the parabola equation, point (x1,y1) and the equation of the tangent line, you should be able to show the conclusion.
    Last edited by chen09; September 24th 2013 at 03:50 PM.
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    Re: Help proving the formula for a tangent line of a parabola

    Thanks very much, guys!

    Now I'm trying to find the x-intercept. Using the equation for a tangent line given in the original post,

    0 = 2p(x + x1)

    2px + 2px1 = 0
    2px1 = -2px
    x1 = -x

    So apparently, the coordinates for the x-intercept are (-x, 0), but the drawing I made makes this hard to believe. I know this is incredibly elementary algebra and whatnot, but I feel like I made a silly mistake somewhere.
    Last edited by RubberDucky; September 24th 2013 at 04:17 PM.
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    Re: Help proving the formula for a tangent line of a parabola

    Know that  y_1=\frac{2px_1}{y_1}+b
    and  (y_1)^2=4px_1
    Derive the result.
    Last edited by chen09; September 24th 2013 at 05:02 PM.
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  6. #6
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    Re: Help proving the formula for a tangent line of a parabola

    The x-intercept of the tangent line, which as shown, has the equation

    y1y = 2p(x+x1)
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    Re: Help proving the formula for a tangent line of a parabola

    Quote Originally Posted by RubberDucky View Post
    The x-intercept of the tangent line, which as shown, has the equation

    y1y = 2p(x+x1)
    use the point slope formula: y - y1 = (2p/y)(x - x1)
    Multiply both sides by y and rearrange, you will obtain the answer to your problem
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  8. #8
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    Re: Help proving the formula for a tangent line of a parabola

    Quote Originally Posted by votan View Post
    use the point slope formula: y - y1 = (2p/y)(x - x1)
    Multiply both sides by y and rearrange, you will obtain the answer to your problem
    I get

    y^2 = 2px - 2px1
    y^2 - 2px = -2px1
    x1 = -(y^2 - 2px)/2p

    Does that look reasonable? From the wording on my assignment, I'm supposed to be able to easily draw a tangent line to a general point P(x1, y1) using the x-intercept, but that's proving challenging with the derived formulas, and I'm just paranoid that I've done something slightly wrong somewhere.
    Last edited by RubberDucky; September 24th 2013 at 06:31 PM.
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    Re: Help proving the formula for a tangent line of a parabola

    Quote Originally Posted by RubberDucky View Post
    I get

    y^2 = 2px - 2px1
    y^2 - 2px = -2px1
    x1 = -(y^2 - 2px)/2p

    Does that look reasonable? From the wording on my assignment, I'm supposed to be able to easily draw a tangent line to a general point P(x1, y1) using the x-intercept, but that's proving challenging with the derived formulas, and I'm just paranoid that I've done something slightly wrong somewhere.

    y^2 = 2px - 2px1 <---- what happened to y*y1 term on the left side?

    Help proving the formula for a tangent line of a parabola-untitled2.gif
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  10. #10
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    Re: Help proving the formula for a tangent line of a parabola

    Quote Originally Posted by votan View Post
    y^2 = 2px - 2px1 <---- what happened to y*y1 term on the left side?

    Click image for larger version. 

Name:	untitled2.gif 
Views:	6 
Size:	8.5 KB 
ID:	29292
    But at the x-intercept, y1 should be zero, right?

    And we've just arrived back at the original equation for the tangent line, so I don't see the relevance. Shouldn't I be solving for x1?
    Last edited by RubberDucky; September 25th 2013 at 02:59 AM.
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    Re: Help proving the formula for a tangent line of a parabola

    Quote Originally Posted by RubberDucky View Post
    But at the x-intercept, y1 should be zero, right?

    And we've just arrived back at the original equation for the tangent line, so I don't see the relevance. Shouldn't I be solving for x1?
    y1 is the ordinate of P. Why you wanted to set it to 0? The statement of the problem you posted does not say y1 = 0.
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