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Math Help - coordinate geometry

  1. #1
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    coordinate geometry

    Show that the tangent to the ellipse ax^2+by^2=1 at the point (h,k) has equation ahx+bky=1

    Hence, deduce that the chord of contact of tangents from the point (m,n) to the ellipse ax^2+by^2=1 has equation amx+bny=1

    I can do the 2 questions above.


    Show that for all values of t, the chord of contact of tangents from the point (2t, 1-t) to the ellipse ax^2+by^2=1 passes through a fixed point and determine the point of this coordinates.

    I am only stuck with this.
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  2. #2
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    Quote Originally Posted by hooke View Post
    Show that the tangent to the ellipse ax^2+by^2=1 at the point (h,k) has equation ahx+bky=1

    Hence, deduce that the chord of contact of tangents from the point (m,n) to the ellipse ax^2+by^2=1 has equation amx+bny=1

    I can do the 2 questions above.


    Show that for all values of t, the chord of contact of tangents from the point (2t, 1-t) to the ellipse ax^2+by^2=1 passes through a fixed point and determine the point of this coordinates.

    I am only stuck with this.
    Put m = 2t and n = 1 t in the equation of the chord: 2tax + (1-t)by = 1. Write that as t(2ax-by) = 1-by. If that holds for all t then 2ax-by = 1-by = 0. Solve those equations to find the values of x and y for the fixed point.
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