? 發問於 科學及數學數學 · 1 十年 前

Maths(Coordinate Treatment of Simple Locus Problems)

The equation of line L is x+3y-2+k(x-y-6)=0

(a)Express the slope of L interms of k.

(b)FInd the value of k if L is parallel to x+2y-4=0.

(c)If L passes through a fixed point P for any value of k,find the coorsinates of P.

2 個解答

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  • WM
    Lv 6
    1 十年 前
    最佳解答

    (a) L: x+3y-2+k(x-y-6) = 0

    x+3y-2+k(x-y-6) = 0

    x + 3y - 2 + kx - ky - 6k = 0

    ky - 3y = kx + x - 6k - 2

    (k - 3)y = (k + 1)x + - 2(3k +1) =0

    y = [(k + 1)/(k - 3)]x - [2(3k +1)/(k - 3)]

    therefore, slope of L = (k + 1)/(k - 3)

    (b) the line x+2y-4=0 can be expressed as:

    2y = -x + 4

    y = -(1/2)x + (4/2)

    therefore slope of the line x+2y-4=0 is -1/2

    if L is parallel to the line x+2y-4=0

    slope of L = -1/2

    (k + 1)/(k - 3) = -1/2

    2k + 2 = -k + 3

    3k = 1

    k = 1/3

    (c) L can be written as (k+ 1)x + (3 -k)y - 2(3k + 1) = 0

    Let the coordinates of P be (a, b), and substitue (a,b) into L, it becomes

    (k+ 1)a + (3 - k)b - 2(3k + 1) = 0 .................(1)

    L passes through a fixed point P for any value of k, so

    put k = -1, (1) becomes

    (-1 + 1)a + [3-(-1)]b - 2[3(-1) + 1] = 0

    4b + 4 = 0

    b = -1

    put k = 3, (1) becomes

    (3+ 1)a + (3 - 3)b - 2[3(3) + 1] = 0

    4a - 20 = 0

    a = 5

    therefore, the coordinates of P is (5, -1)

  • 1 十年 前

    The equation of line L is x+3y-2+k(x-y-6)=0

    (a)Express the slope of L interms of k.

    由L:x+3y-2+k(x-y-6)=0

    x+3y-2+kx-ky-6k=0

    ky-3y=kx+x-(6k+2)

    (k-3)y=(k+1)x-(6k+2)

    y=[(k+1)/(k-3)]x-(6k+2)/(k-3)

    即L的斜率為(k+1)/(k-3)

    (b)FInd the value of k if L is parallel to x+2y-4=0.

    ∵L is parallel to x+2y-4=0

    x+2y-4=0的斜率為-1/2

    即(k+1)/(k-3)=-1/2

    2k+2=-k+3

    3k=1

    ∴k=1/3

    (c)If L passes through a fixed point P for any value of k,find the coorsinates of P

    由L : x+3y-2+k(x-y-6)=0與k無關得

    {x+3y-2=0---------(1)

    {x-y-6=0------------(2)

    (1)-(2)

    4y+4=0

    y=-1

    代入(2)

    x+1-6=0

    x=5

    ∴P的坐標為(5,-1)

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