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Circle and Chords intersected by x-axis

  1. Nov 8, 2011 #1

    AGNuke

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    Let a circle be given by 2x (x-a) + y(2y-b) = 0; (a≠0, b≠0).

    Find the condition on a and b if two chords, each bisected by the x-axis, can be drawn to the circle from (a, b/2)


    My attempt in this question is not quite relevant at this moment. I just found that (a,b/2) will lie on circle and the equation of chord being b(x-p) - 2y(a-p) = 0; where (p,0) is the midpoint of a chord.

    Further than that, I have no idea what to do... :|
     
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  3. Nov 8, 2011 #2

    SammyS

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    If the x-axis bisects the chord, what is the y-coordinate of both of the other endpoints?
     
  4. Nov 9, 2011 #3

    AGNuke

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    One point of the chord, which is present on the circle as well is (a, b/2), which is given, other than that, no relevant info is given.

    I'm trying to solve the question in due time and I hope I can solve this question in my second attempt, using entirely new approach.
     
  5. Nov 9, 2011 #4

    SammyS

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    My point is: The y coordinate of the point midway between (x1, y1) and (x2, y2) is given by [itex]\displaystyle\frac{y_1+y_2}{2}\,.[/itex]

    If [itex]\displaystyle\frac{y_1+y_2}{2}=0[/itex] (The y coordinate for any point on the x-axis is zero.), then if y1 = b/2, what is y2 ?
     
  6. Nov 10, 2011 #5

    AGNuke

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    That's the problem, it is not given. But from section formula, it is obvious that it is -b/2
     
  7. Nov 10, 2011 #6

    HallsofIvy

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    You don't need any formulas! If [itex](y_1+ y_2)/2= 0[/itex] and [itex]y_1= b/2[/itex] then [itex](b/2+ y_2)/2= 0[/itex] so, multiplying through by 2, [itex]b/2+ y_2= 0[/itex] and then [itex]y_2= -b/2.[/itex]
     
  8. Nov 10, 2011 #7

    AGNuke

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    But isn't gif.latex?\frac{y_{1}+y_{2}}{2}=0.gif is a Section formula itself?
     
  9. Nov 10, 2011 #8

    HallsofIvy

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    I don't know what "section" you mean. I would take "[itex](y_1+ y_2)/2= 0[/itex]" as coming from the given information that the x-axis bisects the chord.
     
  10. Nov 10, 2011 #9

    AGNuke

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    Section Formula? To find out the point dividing the distance between two co-ordinates in a fixed ratio either internally or externally.
     
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