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Augustus asked in 科學及數學數學 · 1 decade ago

Question (Circles)

Consider S: x2 + y2 + (k-8)x + (2k-10)y + (1-4k) = 0, where k is a real number. As k varies, S represents a family of circles passing through two fixed points. Find the coordinates of these two points.

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  • 1 decade ago
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    S: x2 + y2 + (k-8)x + (2k-10)y + (1-4k) = 0

    From the condition

    Sub k=8

    x2 + y2 + 6y = 31

    Sub k=5

    x2 + y2 -3x = 19

    These two circle passing through two fixed points

    So 31-6y=3x+19

    3x+6y-12=0

    x+2y-4=0

    x=4-2y

    Sub into x2 + y2 + 6y = 31

    y^2-2y-3=0

    y=-1 or 3

    x=6 or -2

    The coordinates of these two points are (6,-1) and (-2,3) respectively

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