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wing hang 發問於 科學及數學數學 · 1 十年前

coordinate geometry

Let y=mx+c be a chord of the ellipse(x^2/a^2)+(y^2/b^2)=1

a) If the chord subtends a right angle at the point (a,0), show that (a^2+b^2)c^2+2a^3mc+a^2(a^2-b^2)m^2=0

b)Hence show that the chord passes through a fixed point on the x-axis

2 個解答

評分
  • 1 十年前
    最愛解答

    (a) With reference to the diagram below:

    圖片參考:http://i117.photobucket.com/albums/o61/billy_hywun...

    (b) The x-intercept of the chord is given by substituting y = 0 in its equation:

    0 = mx + c

    x = -c/m

    Therefore, it can be found from the result of (a) as follows:

    圖片參考:http://i117.photobucket.com/albums/o61/billy_hywun...

    Since a and b are fixed, the x-intercept is also fixed and hence the result comes.

    資料來源: My Maths knowledge
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  • wy
    Lv 7
    1 十年前

    (x^2/a^2) + (y^2/b^2) = 1.......(1)

    y = mx +c ......................(2)

    Substitute y of (2) into (1), we get a quadratic equation in x. Let the roots of the equation by h and k. It can be found that :

    hk = a^2(c^2-b^2)/(b^2 + a^2m^2)...........(3) and

    h + k = -2mca^2/(b^2 + a^2m^2).........(4)

    Also, h and k are the x-coordinates of the intersecting points of the chord with the ellipse.

    Since these 2 points make a right angle with (a,0), then

    slope of point h with (a,0) x slope of point k with (a,0) = -1. That is

    [y-coordinates of h/(h-a)][y-coordinates of k/(k-a)] = -1

    y- coordinate of h x y-coordinate of k = -(h-a)(k-a)

    (mh +c)(mk+c) = -(h-a)(k-a)

    m^2hk + c^2 + mc(h + k) = -hk - a^2 + a(h+k)

    (1+ m^2)(hk) + c^2 + a^2 + (mc - a)(h + k) = 0

    Substitute the value of (3) and (4) into this equation and after simplification work, you will come to the result.

    b)

    Since y= mx+ c

    For y = 0, x = -c/m......(5)

    Using the result of (a) part and dividing all terms by m^2, we get

    (a^2 + b^2)(c/m)^2 + 2a^3(c/m) + a^2(a^2 - b^2) = 0 which is a quadratic equation in c/m.

    Since a and b are constants, the roots of c/m are also constants, so x in (5) is a constant. That means the chord passes through a fixed point on the x-axis.

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