Simple coordinate geomety problem

  1. I'm not sure where to post this. So I'm posting in the General Math section.This is a simple coordinate geometry problem: We have to find the equation of line(s) passing through the point (7,17) and having a distance of 6 units from the point(1,9).

    Now I'm posting my approach:
    the equation of line passing through(7,17) and having slope m is
    y-17=m(x-7)
    or, mx-y-7m-17=0
    now the line have a distance of 6 units from point(1,9).
    so [itex]\frac{m-9-7m+17}{\sqrt{m^2 +1}}[/itex]=[itex]\pm[/itex]6
    or,(6m-8)2 =36(m2 +1)
    or, m=7/24
    so the equation of line becomes 7x-24y+359 =0
    This way we get only line.But if you draw it in a graph paper,you'll see that there should be another line which is x-7=0 which is parallel to the y-axis.(draw a circle of radius 6 from (1,9) and then draw tangent from (7,17) to the circle).
    I think we can't get the second one because it has undefined slope.
    My question is how can I get the second line without plotting in graph?
     
    Last edited: Feb 18, 2014
  2. jcsd
  3. tiny-tim

    tiny-tim 26,054
    Science Advisor
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    hi silent_hunter! :smile:

    (try using the X2 button just above the Reply box :wink:)

    you lost half the solutions when you took the square-root of this line …
    you should have put the intermediate step (6m-8) = ±36(m2 +1) :wink:
     
  4. thanks for your reply :smile:
    (6m-8)2 =36(m2 +1)
    in that step I squared both sides of the equation ,so the ± sign should go away.
     
  5. tiny-tim

    tiny-tim 26,054
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    sorry, i got that wrong :redface:

    this is where you missed a solution …
    … you missed out m = ∞ ! :smile:
     
  6. sorry I made a typing mistake. :blushing:
    Its mx-y-7m-17=0 but I didn't get what you said. Would you please elaborate?Thanks.
     
  7. tiny-tim

    tiny-tim 26,054
    Science Advisor
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    one of the two lines is x = 7, isn't it?

    (with slope ∞)

    that doesn't come up for any m in mx-y-7m-17=0 :wink:
     
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  8. if you think, (y-9)^2+(x-1)^2=36 then

    if you think of it as (y-9)^2 + (x-1)^2 = 36

    then:

    y=cos(3(x-1))+15.91 or so should hit twice.

    I'll bet some kind of absolute value function would hit three times.
     
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