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Math Help - Geometrical Properties of Ellipses

  1. #1
    Junior Member
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    Exclamation Geometrical Properties of Ellipses

    Hi.

    Could someone please help me with these ellipse problems?

    1. If M and M' are the points where the tangent at P on the ellipse meets the tangents at the ends of the major axis, then MM' subtends a right angle at either focus. Show that the line y = mx + c touches the ellipse x^2/a^2 + y^2/b^2 = 1 when c = +/- sqrt[a^2m^2+b^2]

    2. The normal to the ellipse x^2/a^2 + y^2/b^2 = 1 at P(x1, y1) meets the x-axis in N and the Y-axis in G. Prove PN/NG = (1-e^2)/e^2.

    3. Show that the semi-latus rectum of the ellipse (acosx, bsiny) is of length b^2/a.

    4. The extremities of any diameter of an ellipse are L, L' and P is any point on the ellipse. Show that the product of the gradients of the chords LP and L'P are constant.
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  2. #2
    MHF Contributor
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    Hi

    The line y = mx+c touches the ellipse \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 when

    \frac{x^2}{a^2} + \frac{\left(mx+c\right)^2}{b^2} = 1 has 1 solution

    \left(\frac{1}{a^2}+\frac{m^2}{b^2}\right)x^2 + \frac{2mc}{b^2} x + \frac{c^2}{b^2} - 1 = 0 has 1 solution

    therefore when the discriminant is equal to 0 which gives c^2 = a^2m^2 + b^2 after simplification
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