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Math Help - showing that 3 normals can be drawn to parabola

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    Exclamation showing that 3 normals can be drawn to parabola

    A parabola has the parametric equations x = 2at and y = at^2. The normal to the parabola when t= p is  x + py = 2ap + ap^3. Show that through  (x_1, y_1) it is possible to draw up to three normals to the parabola



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  2. #2
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    Geometry+Algebra = Analytical Geometry

    Quote Originally Posted by differentiate View Post
    A parabola has the parametric equations x = 2at and y = at^2. The normal to the parabola when t= p is  x + py = 2ap + ap^3. Show that through  (x_1, y_1) it is possible to draw up to three normals to the parabola



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    Geometry: The point  (x_1, y_1) lies on the line  x + py = 2ap + ap^3. for some p.
    Algebra: Substitute  (x_1, y_1) in  x + py = 2ap + ap^3.

    Geometry: To prove that it is possible to draw up to three normals to the parabola x^2 = 4ay of the form  x + py = 2ap + ap^3. .
    Algebra: To prove that it is possible that there are up to three solutions p_1,p_2,p_3 for the equation  x_1 + py_1 = 2ap + ap^3. .

    Now that we have formulated the problem algebraically, can you tell why "there are up to three solutions p_1,p_2,p_3 for the equation  x_1 + py_1 = 2ap + ap^3. "?
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