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DLS

The equation of circle which passes through (1,0),(0,1) & has its Radius as smal as possible is?

  • one year ago
  • one year ago

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  1. ParthKohli
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    There's only one unique circle passing through (1,0) and (0,1) right?

    • one year ago
  2. rajathsbhat
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    No.

    • one year ago
  3. DLS
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    i dont think so

    • one year ago
  4. ParthKohli
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    Thanks for clearing my doubts. That's what I was trying out

    • one year ago
  5. ParthKohli
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    May I have a counterexample BTW?

    • one year ago
  6. DLS
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    radius is your choice

    • one year ago
  7. rajathsbhat
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    |dw:1357382125336:dw|

    • one year ago
  8. ParthKohli
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    Thanks.

    • one year ago
  9. ParthKohli
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    Two circles that intersect, right. =)

    • one year ago
  10. rajathsbhat
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    the smallest circle is the one that has the line connecting the two points as the diameter.

    • one year ago
  11. DLS
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    wait what did you say :?

    • one year ago
  12. DLS
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    and why are there 2 circles

    • one year ago
  13. ParthKohli
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    Radius of \(1\)? |dw:1357382192536:dw|

    • one year ago
  14. ParthKohli
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    That's just the unit circle.

    • one year ago
  15. rajathsbhat
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    all other circles have this line as their chord which means that their diameters are longer that this line. An extreme example:|dw:1357382299966:dw|

    • one year ago
  16. ParthKohli
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    If you draw a line segment, you'd get a segment with length \(\sqrt 2\). That is the smallest segment (and the only) possible. Now you get a triangle. Draw a circle around that triangle and that would be \(x^2 + y^2 = 1\).|dw:1357382531256:dw|

    • one year ago
  17. rajathsbhat
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    why all this?

    • one year ago
  18. ParthKohli
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    \[r^2 = h^2 + (k - 1)^2 = (h - 1)^2 + k^2\]

    • one year ago
  19. DLS
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    I just want to know why should they be diameterically opp..to satisfy the eq.nothing else :/

    • one year ago
  20. ParthKohli
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    \[h^2 + k^2 -2k+1=h^2 - 2h + 1 + k^2\]\[-2h + 1 = -2k +1 \iff h =k\]So \((0,0)\)?

    • one year ago
  21. ParthKohli
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    \((0,0)\) is the center.

    • one year ago
  22. DLS
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    why u no tell jitna pucha :(

    • one year ago
  23. ParthKohli
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    1 answer

    • one year ago
  24. ParthKohli
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    \[x^2 + y^2 = 1\]

    • one year ago
  25. ParthKohli
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    Pakka yahi hai answer.

    • one year ago
  26. DLS
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    lol no way

    • one year ago
  27. ParthKohli
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    Why this Kolaveri

    • one year ago
  28. rajathsbhat
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    the concept is pretty straightforward. Look at this diagram|dw:1357382709667:dw| The line you see is the line connecting (1,0) and (0,1) And i've drawn two circles that pass though the two points. The line drawn is a chord for the bigger circle which means that it's diameter is bigger that the line, ok? But look at the other circle which has the line as the diameter. It's obviously smalller than the other circle but also smaller than any other circle that can be drawn though those two points (again because chords are smaller than diameters)|dw:1357383015575:dw| So, the circle you're looking for is the circle that has that line for it's diameter. Simple.

    • one year ago
  29. DLS
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    not so clear but yeah somewhat..

    • one year ago
  30. ParthKohli
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    another way to think is that you're looking for the smallest right triangle \(ABC\) where \(\angle B\) is the right angle as well as the center of the circle and the circle passes through \(A\) and \(C\).

    • one year ago
  31. DLS
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    how can (1,0) and (0,1) be diameterically opp but :O

    • one year ago
  32. ParthKohli
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    I didn't say they were diametrically opposite =/

    • one year ago
  33. DLS
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    :S

    • one year ago
  34. rajathsbhat
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    but i did.

    • one year ago
  35. rajathsbhat
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    they have to be.

    • one year ago
  36. ParthKohli
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    They are diametrically perpendicular...

    • one year ago
  37. DLS
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    how on eart h:/

    • one year ago
  38. rajathsbhat
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    No parth. I don't even know what that means... But i'll draw on paper and show you what I mean.

    • one year ago
  39. ParthKohli
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    |dw:1357383885601:dw|

    • one year ago
  40. DLS
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    kya yaa dimaag ka dahi mat karo :P aur bhi q hai!!

    • one year ago
  41. rajathsbhat
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    that's absolutely wrong. I can draw a smaller circle!|dw:1357384070442:dw| you see the point!? this right here the smallest circle.

    • one year ago
  42. ParthKohli
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    Oh, never thought of that =P

    • one year ago
  43. DLS
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    OH ! !

    • one year ago
  44. ParthKohli
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    So then it's\[(x - 0.5)^2 + (y - 0.5)^2 = 0.25\]?

    • one year ago
  45. DLS
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    no

    • one year ago
  46. ParthKohli
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    So that's the trick. You make them diametrically opposite

    • one year ago
  47. ParthKohli
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    @rajathsbhat Yaar chaa gaye =P

    • one year ago
  48. DLS
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    (x-x1)(x-x2)+(y-y1)(y-y2)=0 hoti hai equation diameterically op points ki :D

    • one year ago
  49. rajathsbhat
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    that's right^

    • one year ago
  50. DLS
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    aaree yaar aage bado :P

    • one year ago
  51. ParthKohli
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    But we want the equation of the circle!

    • one year ago
  52. DLS
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    yeah wo circle ki he hai

    • one year ago
  53. ParthKohli
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    Achcha okay! Point noted

    • one year ago
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