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Math Help - locus - ellipse

  1. #1
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    locus - ellipse

    p143 q9 c

    I have problem in (c). I get the answer 2x+y = 0 but the book says it is 2x+3y = 0
    question:
    consider an ellipse E:  \frac {x^2}9 + \frac {y^2}4 = 1
    and a line L :  2x-3x+6=0
    and D: y=mx+c is a line parallel to L.
    a) if D cuts E at Q(x_1 , y_1) and R(x_2 , y_2) , show that x_1 and x_2 are the roots of the equation
    8x^2 +12cx + 9c^2- 36=0
    b) find the range of values of c if E and D cut at (i) 2 points / (ii) 1 point
    c) M is the mid - point of QR. Find the equation of the locus M as c varies.

    here's my working in (c)
    D: y = - \frac 2 3 x + c
    x_1 + x_2 = - \frac {12c}8 = - \frac {3c}2
    x = - \frac {3c}4
    y_1 +y_2 = - \frac 2 3 (x_1 +x_2) + 2x = 3c
    y= \frac 3 2 c = -2 x
    2x+y = 0
    but answer says it's 2x+3y=0. what's wrong??? thanks!
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  2. #2
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    Quote Originally Posted by afeasfaerw23231233 View Post
    p143 q9 c

    I have problem in (c). I get the answer 2x+y = 0 but the book says it is 2x+3y = 0
    question:
    consider an ellipse E:  \frac {x^2}9 + \frac {y^2}4 = 1
    and a line L :  2x-3x+6=0 I assume that you mean: 2x-3y+6=0
    and D: y=mx+c is a line parallel to L.
    a) if D cuts E at Q(x_1 , y_1) and R(x_2 , y_2) , show that x_1 and x_2 are the roots of the equation
    8x^2 +12cx + 9c^2- 36=0
    b) find the range of values of c if E and D cut at (i) 2 points / (ii) 1 point
    c) M is the mid - point of QR. Find the equation of the locus M as c varies.

    here's my working in (c)
    D: y = - \frac 2 3 x + c
    x_1 + x_2 = - \frac {12c}8 = - \frac {3c}2
    x = - \frac {3c}4
    y_1 +y_2 = - \frac 2 3 (x_1 +x_2) + 2x = 3c
    y= \frac 3 2 c = -2 x
    2x+y = 0
    but answer says it's 2x+3y=0. what's wrong??? thanks!
    If you have 2x-3y+6=0~\iff~ 2x+6=3y~\iff~y=\frac23 x +2

    So you have used the wrong slope with D
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  3. #3
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    oh, no! i am so careless. thanks
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