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Math Help - Precalculus Parabola Question

  1. #1
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    Precalculus Parabola Question

    Prove that for a parabola of the form x2 = 4py, for ANY focal chord, lines passing through either endpoint of the focal chord and the vertex will pass through the directrix at the foot of the perpendicular line that can be dropped from the opposite focal chord endpoint to the directrix.


    Your proof should use variables to show that it holds for any choice of focal chord.

    I know that for any focal chord of any length, one endpoint's coordinates are (x,y) and (-4p^2/x,p^2,y). But how can I use that to solve this question (assuming I even need it)?



    Basically, I need to prove that the red and blue line both hit the directrix at the same point, the blue line passes through one endpoint of the (green) focal chord and the vertex, the red line is perpendicular to the green focal chord.
    Last edited by mathstudent1235; March 2nd 2014 at 01:22 PM.
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    Re: Precalculus Parabola Question

    Quote Originally Posted by mathstudent1235 View Post
    Prove that for a parabola of the form x2 = 4py, for ANY focal chord, lines passing through either endpoint of the focal chord and the vertex will pass through the directrix at the foot of the perpendicular line that can be dropped from the opposite focal chord endpoint to the directrix.


    Your proof should use variables to show that it holds for any choice of focal chord.

    I know that for any focal chord of any length, one endpoint's coordinates are (x,y) and (-4p^2/x,p^2,y). But how can I use that to solve this question (assuming I even need it)?



    Basically, I need to prove that the red and blue line both hit the directrix at the same point, the blue line passes through one endpoint of the (green) focal chord and the vertex, the red line is perpendicular to the green focal chord.
    Here is how I would start it. I'd name things.

    $\left(a, \dfrac{a^2}{4p}\right)\ is\ the\ left\ endpoint\ of\ the\ focal\ chord.$

    $\left(b, \dfrac{b^2}{4p}\right)\ is\ the\ right\ endpoint\ of\ the\ focal\ chord.$

    Now I would start writing down information implied by the problem. For example, what are the co-ordinates of the focus? So what is the equation of the focal chord? What is the equation of the line joining the right endpoint to the vertex? Where does that line intersect the directrix? What is the equation of the line from that intersection to the left endpoint? What is the relationship between the slopes of two perpendicular lines. At this point I do not know which information may be relevant and which irrelevant. I just want to see it all and not forget any of it. Once I have the pieces of the puzzle in front of me, then I can begin to think.
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    Re: Precalculus Parabola Question

    Quote Originally Posted by JeffM View Post
    Here is how I would start it. I'd name things.

    $\left(a, \dfrac{a^2}{4p}\right)\ is\ the\ left\ endpoint\ of\ the\ focal\ chord.$

    $\left(b, \dfrac{b^2}{4p}\right)\ is\ the\ right\ endpoint\ of\ the\ focal\ chord.$

    Now I would start writing down information implied by the problem. For example, what are the co-ordinates of the focus? So what is the equation of the focal chord? What is the equation of the line joining the right endpoint to the vertex? Where does that line intersect the directrix? What is the equation of the line from that intersection to the left endpoint? What is the relationship between the slopes of two perpendicular lines. At this point I do not know which information may be relevant and which irrelevant. I just want to see it all and not forget any of it. Once I have the pieces of the puzzle in front of me, then I can begin to think.
    I have everything but the red part. I tried using the perpendicular thing which eventually would lead to a system of equations that resulted in y=-p as the y-answer (as that is the directrix equation). I also tried the distance formula, but that failed. How should I proceed? I have no idea what is relevant and what isn't.
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  4. #4
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    Re: Precalculus Parabola Question

    if you have a vertical parabola centered at x=0, w/directrix line y=-p, then your focus is at (0,p)
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    Re: Precalculus Parabola Question

    Quote Originally Posted by mathstudent1235 View Post
    I have everything but the red part. I tried using the perpendicular thing which eventually would lead to a system of equations that resulted in y=-p as the y-answer (as that is the directrix equation). I also tried the distance formula, but that failed. How should I proceed? I have no idea what is relevant and what isn't.
    If you already have everything, then what is the ratio of the slope of the focal chord and the line joining one endpoint and the intersection of the directrix with the line running from the other endpoint through the origin? What does that ratio mean?
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