## Callisto 2 years ago Haven't done circle for long :( In the figure, a circle S: $$x^2+y^2+6x-2y=0$$ cuts the axes at the origin O, A and C. A point B lies on the circle S so that the length of BC is $$2 \sqrt{5}$$ units. BC is produced to cut the x-axis at D. Find the coordinates of B

1. Callisto

|dw:1356781451262:dw|

2. Callisto

brb

3. experimentX

solve these equations for $$x_1, y_1$$, $x_1^2 + y_1^2 + 6x_1−2y_1=0,\\ x_2^2 + y_2^2 + 6x_2 - 2y_1 = 0, x_2=0, y_2 \neq 0 \\ (x_1-x_2)^2 + (y_1-y_2)^2 = 20$

4. Callisto

May I know what (x2, y2) is??

5. experimentX

x_2, y_2 are the coordinates of C

6. experimentX

probably you would end up with two sets of point, choose the the point where y>0

7. experimentX

sorry ... this equation .. change y1 to y2 $x_2^2 + y_2^2 + 6x_2 - 2y_1 = 0, x_2=0, y_2 \neq 0 \\$

8. Callisto

Algebra matters.

9. Callisto

C = (0, 2)

10. experimentX

yes!! sorry i was careless ... did you find out answer?

11. Callisto

$$x^2 + y^2 +6x - 2y =0$$ ---(1) $$x^2+(y-2)^2 =20$$ => $$x^2 + y^2 -4y+4 =20$$ ---(2)

12. Callisto

he?

13. Callisto

Do you mean I knew the answer?

14. Callisto

I knew the answer, yes. I'm looking for a way to get the answer. I tried once (more than once) but I thought I was making it too complicated. I just wanted to find a simpler way so that my sister can do it easily.

15. hartnn

co-ordinates of A can also be found similarly.

16. Callisto

A = (-6, 0)

17. Callisto

D = (4, 0)

18. hartnn

then u know equation of line BD can't u use that to find co-ordinates of B ?

19. hartnn

solving equation of line and circle simultaneously.

20. hartnn

are u sure about D =(4,0)??

21. Callisto

equation of line BD: $\frac{y-2}{x} = \frac{2}{-4}$$x = 4-2y$ $(4-2y)^2 + y^2 +6(4-2y) - 2y =0$ $5y^2 - 30y + 40=0$$y=4, 2$When y=4, x = 4-2(4) = -4 When y=2, x=4-2(2) =0 (rejected) So, B= (-4, 4) Yay! Much simpler :) Thanks!!! @hartnn Yes! Yes! D = (4,0)

22. hartnn

oh, i somehow got 36 instead of 40, thats why i was stuck.....

23. Callisto

Correction: o.O Did I make a mistake? :S

24. hartnn

no, u are correct.

25. Callisto

Okay, thanks again! I didn't thought of the line. I'm just too stupid :(

26. hartnn

happens! i just did a silly mistake of 6*4 =20 :P

27. shubhamsrg

did anyone notice : |dw:1356789155290:dw|

28. shubhamsrg

where thats the center we know co-ordinates of C let center be O hence we know slope of OB slope given, distance give, we can find co-ordinate! B)

29. hartnn

how do we know slope of OB?

30. hartnn

ohh..

31. hartnn

as we know slope of OC

32. shubhamsrg

yep..exactly..

33. hartnn

given slope of OB and O, we can find B, no need of any distance..

34. Callisto

Not so relevant. |dw:1356789503661:dw|

35. shubhamsrg

ohh yes!! COA are collinear !

36. hartnn

how does that make it any simpler ?

37. shubhamsrg

i dont know,,just a good fact to know! :D

38. Callisto

''Not so relevant.'' :P

39. hartnn

:P ok ..

40. hartnn

u got 2 more easy methods ;)

41. Callisto

Yes! Yes! Thanks! Thanks!!