DLS 2 years ago The equation of circle which passes through (1,0),(0,1) & has its Radius as smal as possible is?

1. ParthKohli

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

2. rajathsbhat

No.

3. DLS

i dont think so

4. ParthKohli

Thanks for clearing my doubts. That's what I was trying out

5. ParthKohli

May I have a counterexample BTW?

6. DLS

7. rajathsbhat

|dw:1357382125336:dw|

8. ParthKohli

Thanks.

9. ParthKohli

Two circles that intersect, right. =)

10. rajathsbhat

the smallest circle is the one that has the line connecting the two points as the diameter.

11. DLS

wait what did you say :?

12. DLS

and why are there 2 circles

13. ParthKohli

Radius of $$1$$? |dw:1357382192536:dw|

14. ParthKohli

That's just the unit circle.

15. rajathsbhat

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

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

why all this?

18. ParthKohli

$r^2 = h^2 + (k - 1)^2 = (h - 1)^2 + k^2$

19. DLS

I just want to know why should they be diameterically opp..to satisfy the eq.nothing else :/

20. ParthKohli

$h^2 + k^2 -2k+1=h^2 - 2h + 1 + k^2$$-2h + 1 = -2k +1 \iff h =k$So $$(0,0)$$?

21. ParthKohli

$$(0,0)$$ is the center.

22. DLS

why u no tell jitna pucha :(

23. ParthKohli

24. ParthKohli

$x^2 + y^2 = 1$

25. ParthKohli

26. DLS

lol no way

27. ParthKohli

Why this Kolaveri

28. rajathsbhat

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

not so clear but yeah somewhat..

30. ParthKohli

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

how can (1,0) and (0,1) be diameterically opp but :O

32. ParthKohli

I didn't say they were diametrically opposite =/

33. DLS

:S

34. rajathsbhat

but i did.

35. rajathsbhat

they have to be.

36. ParthKohli

They are diametrically perpendicular...

37. DLS

how on eart h:/

38. rajathsbhat

No parth. I don't even know what that means... But i'll draw on paper and show you what I mean.

39. ParthKohli

|dw:1357383885601:dw|

40. DLS

kya yaa dimaag ka dahi mat karo :P aur bhi q hai!!

41. rajathsbhat

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

Oh, never thought of that =P

43. DLS

OH ! !

44. ParthKohli

So then it's$(x - 0.5)^2 + (y - 0.5)^2 = 0.25$?

45. DLS

no

46. ParthKohli

So that's the trick. You make them diametrically opposite

47. ParthKohli

@rajathsbhat Yaar chaa gaye =P

48. DLS

(x-x1)(x-x2)+(y-y1)(y-y2)=0 hoti hai equation diameterically op points ki :D

49. rajathsbhat

that's right^

50. DLS

51. ParthKohli

But we want the equation of the circle!

52. DLS

yeah wo circle ki he hai

53. ParthKohli

Achcha okay! Point noted