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DLS

  • 2 years ago

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

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

  2. rajathsbhat
    • 2 years ago
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    No.

  3. DLS
    • 2 years ago
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    i dont think so

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

  5. ParthKohli
    • 2 years ago
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    May I have a counterexample BTW?

  6. DLS
    • 2 years ago
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    radius is your choice

  7. rajathsbhat
    • 2 years ago
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    |dw:1357382125336:dw|

  8. ParthKohli
    • 2 years ago
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    Thanks.

  9. ParthKohli
    • 2 years ago
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    Two circles that intersect, right. =)

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

  11. DLS
    • 2 years ago
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    wait what did you say :?

  12. DLS
    • 2 years ago
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    and why are there 2 circles

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

  14. ParthKohli
    • 2 years ago
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    That's just the unit circle.

  15. rajathsbhat
    • 2 years ago
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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|

  16. ParthKohli
    • 2 years ago
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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|

  17. rajathsbhat
    • 2 years ago
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    why all this?

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

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

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

  21. ParthKohli
    • 2 years ago
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    \((0,0)\) is the center.

  22. DLS
    • 2 years ago
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    why u no tell jitna pucha :(

  23. ParthKohli
    • 2 years ago
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    1 answer

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

  25. ParthKohli
    • 2 years ago
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    Pakka yahi hai answer.

  26. DLS
    • 2 years ago
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    lol no way

  27. ParthKohli
    • 2 years ago
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    Why this Kolaveri

  28. rajathsbhat
    • 2 years ago
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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.

  29. DLS
    • 2 years ago
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    not so clear but yeah somewhat..

  30. ParthKohli
    • 2 years ago
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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 \(C\).

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

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

  33. DLS
    • 2 years ago
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    :S

  34. rajathsbhat
    • 2 years ago
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    but i did.

  35. rajathsbhat
    • 2 years ago
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    they have to be.

  36. ParthKohli
    • 2 years ago
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    They are diametrically perpendicular...

  37. DLS
    • 2 years ago
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    how on eart h:/

  38. rajathsbhat
    • 2 years ago
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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.

  39. ParthKohli
    • 2 years ago
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    |dw:1357383885601:dw|

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

  41. rajathsbhat
    • 2 years ago
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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.

  42. ParthKohli
    • 2 years ago
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    Oh, never thought of that =P

  43. DLS
    • 2 years ago
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    OH ! !

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

  45. DLS
    • 2 years ago
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    no

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

  47. ParthKohli
    • 2 years ago
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    @rajathsbhat Yaar chaa gaye =P

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

  49. rajathsbhat
    • 2 years ago
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    that's right^

  50. DLS
    • 2 years ago
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    aaree yaar aage bado :P

  51. ParthKohli
    • 2 years ago
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    But we want the equation of the circle!

  52. DLS
    • 2 years ago
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    yeah wo circle ki he hai

  53. ParthKohli
    • 2 years ago
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    Achcha okay! Point noted

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