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Math Help - [Coordinate Geometry] Locus

  1. #1
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    [Coordinate Geometry] Locus

    A(-1,4) and B(7,5) are two ends of the diameter of a circle . Find the equation of the locus of point C such that <ACB is a right angle .

    Thanks so much !
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  2. #2
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    Quote Originally Posted by CallMeBin View Post
    A(-1,4) and B(7,5) are two ends of the diameter of a circle . Find the equation of the locus of point C such that <ACB is a right angle .

    Thanks so much !
    Let C be (x, y)
    AC is perpendicular to BC
    So slope of AC*slope of BC = -1.
    Find the slopes of AC and BC and find the locus of C.
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  3. #3
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    So the slope/gradient will be in unknown as C(x,y) ?
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  4. #4
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    Quote Originally Posted by CallMeBin View Post
    A(-1,4) and B(7,5) are two ends of the diameter of a circle . Find the equation of the locus of point C such that <ACB is a right angle .

    Thanks so much !
    hi

    OR

    Use the phythagoras theorem . Let c be (x,y)

    (x+1)^2+(4-y)^2+(7-x)^2+(5-y)^2=65

    then simplify it , you will find the locus of c to be a circle because the point C can be any point on the circumference since the angle in the semicircle is a right angle .
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  5. #5
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    Quote Originally Posted by CallMeBin View Post
    A(-1,4) and B(7,5) are two ends of the diameter of a circle . Find the equation of the locus of point C such that <ACB is a right angle .

    Thanks so much !
    HI all
    I think c will be any point belongs to the circle
    (x-3)^2 + (y - 9/2)^2 = 65
    except (-1 , 4) , (7 , 5 )
    mrmohamed
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  6. #6
    Junior Member mrmohamed's Avatar
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    Quote Originally Posted by mathaddict View Post
    hi

    OR

    Use the phythagoras theorem . Let c be (x,y)

    (x+1)^2+(4-y)^2+(7-x)^2+(5-y)^2=65

    then simplify it , you will find the locus of c to be a circle because the point C can be any point on the circumference since the angle in the semicircle is a right angle .
    sorry mathaddict
    I didnt see your post
    mrmohamed
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  7. #7
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    Thanks . I've done it . But I still don't understand why will it become like that (My way)

    May I ask why = 65 ?
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  8. #8
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    Quote Originally Posted by CallMeBin View Post
    Thanks . I've done it . But I still don't understand why will it become like that (My way)

    May I ask why = 65 ?
    65 is the length of the diameter calculated using the distance formula .

    @Mr Mohammad , no worries
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  9. #9
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    Quote Originally Posted by CallMeBin View Post
    So the slope/gradient will be in unknown as C(x,y) ?
    A(-1,4) and B(7,5)
    Slope of AC = (y-4)/(x+1)
    Slope of BC = (y-5)/(x-7)
    They are perpendicular because ACB is a right angled triangle.So
    (y-4)(y-5)/(x+1)(x-7) = -1.
    After simplification you will get the locus of C.
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  10. #10
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    Thanks everyone !

    This question is so special . The questions given by my teacher all are given the ratio but this one use distance instead of ratio .
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