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Prove that two parabolas intersect orthogonally

Consider two parabolas which have the same focus and the same line as axis, and let these two parabolas have vertices lying on opposite sides of the focus. Prove that the parabolas intersect orthogonally (i.e., their tangent lines are perpendicular at the point of intersection).


One comment

  1. S says:

    In the note in section 4.7, he gave explanation why the two non-vertical lines with slopes having product -1 are perpendicular, so if we prove that the derivatives at the point of intersection have product -1, we are done. To simplify calculations, we can place the coordinate system such that the x axis is the axis of the parabolas, so F=(f,0), N=(n,0), P=(p,0) be a point on the directrix 1, Q=(q,0) be a point on the second directrix, and let X=(x,y) be the common point.

    The point on one parabola is x=(y^2+f^2-p^2)/(2(f-p)) and on the other one is x=(y^2+f^2-q^2)/(2(f-q)). Taking the common point, we get the common y. We then take the derivative of two xs at that point, and see that their product is -1, as needed.

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