A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

anonymous

  • one year ago

Find circle which is tangent to x-axis and path through points (1,-2) and (3,-4). I worked it through analytic geometry and it gives two possible circles!!! and I can not figure how 3 conditions for a circle gives 2 answers or how to draw it geometrically?

  • This Question is Closed
  1. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @ganeshie8

  2. welshfella
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    general equation is (x - a)^2 + ( y - b)^2 = r^2 you can create 2 equations in a , b and r by substituting the 2 given points

  3. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Circle is clearly defined by 3 conditions. How can I draw 2 circles which path through two points and tangent to a line!!

  4. welshfella
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    The circle touches the x -axis so that another point on (x , 0)

  5. welshfella
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    - good question - I'm trying to figure that

  6. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Can you give a picture? that is puzzling me

  7. welshfella
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    |dw:1432910060214:dw|

  8. welshfella
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    lo1 - not a great diagram

  9. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I'm sorry I meant two possible cirlces

  10. welshfella
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    |dw:1432910234711:dw|

  11. welshfella
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    thats worse than the first

  12. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    :) Yeah that is amazing.

  13. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    two circles can have the same common tangent, yes ?

  14. welshfella
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    yes

  15. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    may be forget about x-axis and look at below diagram |dw:1432910495765:dw|

  16. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    both circles meet below 3 conditions : 1, 2) pass through two points (intersection) 3) having that blue line as tangent

  17. welshfella
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    ah - yes the common tangenytis the x-axis

  18. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I got it now thanks

  19. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    your question about "why 3 conditions are not giving me an unique circle" is very interesting, im still thinking of a better explanation

  20. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I don't know. In such cases I just change that thinking. I think the tangent line condition isn't tough enough ((unless the tangent point is given))

  21. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    anyway that for your helping :)

  22. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    hmm the tangent point is not so random, we only have two choices so its still a bit mysterious

  23. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    |dw:1432911213933:dw|

  24. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    But the result is the same tangent line condition isn't unique as you have said.

  25. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    geometrically how do you know there doesn't exist a third circle that meets the given conditions ?

  26. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    |dw:1432911454842:dw|

  27. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    how do i convince more than two circles are never possible given 1) two points 2) tangent line

  28. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    the center isn't on locus of chord made by these 2 points

  29. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I mean when you bisect it and make a perpendicular which is the locus of the center

  30. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    perpendicular bisector of the chord passes through the center but again how do you know the 3 centers are not collinear ?

  31. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    |dw:1432911763458:dw|

  32. welshfella
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    the line joining the points of contact to the centers of the circles are perpendicular to the tangent - can that be used to answer the question?

  33. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    BINGO!!!

  34. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    |dw:1432912228871:dw|

  35. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    hmm idk

  36. welshfella
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    yes - its a puzzle

  37. welshfella
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    |dw:1432912469311:dw|

  38. welshfella
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    why can't there be an other larger circle like the above?

  39. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    for any triangle it has only one unique circle paths through its vertices |dw:1432912656462:dw| so if we moved the point along the locus the lengths won't be equal and if we shifted it up or down, the the distance to 2pionts won't be equal.

  40. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    brilliant!

  41. welshfella
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    i'm afraid i gotta go and leave this interesting discussion ..

  42. myininaya
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    I know you wanted geometric stuff or whatever but I think I have found two answers with a combination of algebra and calculus.

  43. myininaya
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    It is pretty long that way. :p

  44. myininaya
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    \[\text{ our circle has points: } (1,-2);(3,-4);(a,0) \\ \text{ we have horizontal tangent at } (a,0) \\ (a-h)^2+(0-k)^2=r^2 \\ (1-h)^2+(k+2)^2=r^2 \\ (3-h)^2+(k+4)^2=r^2 \\ y'=\frac{h-x}{y-k} \\ y'|_{x=a}=0=\frac{h-a}{0-k} \implies h=a \\ (a-a)^2+k^2=r^2 \text{ so } k=-r \text{ or } k=r \\ \text{ so going with the } k=r \text{ thing we have } \\ (1-h)^2+(r+2)^2=r^2 \\ (3-h)^2+(r+4)^2=r^2 \\ (1-h)^2+4r+4=0 \\ (3-h)^2+8r+16=0 \\ 2(1-h)^2+8=(3-h)^2+16 \\ h^2+2h-15=0 \\ (h+5)(h-3)=0 \\ h=-5 \text{ or } h=3 \] so pluggin some things back in we can find r then k \[h=-5 \\ a=-5 \\ (1+5)^2+4r+4=0 \\ r=10 \\ k=10,-10\] \[h=3 \\ a=3 \\ (1-3)^2+4r+4=0 \\ r=2 \\ k=2 ,-2\] some of these solutions need to be checked you will see only 2 of the 4 will work and like i said I know this isn't the approach you wanted but I thought it would be nice to try another approach

  45. welshfella
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    looks good to me

  46. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.