## Callisto Group Title 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 one year ago one year ago

1. Callisto Group Title

|dw:1356781451262:dw|

2. Callisto Group Title

brb

3. experimentX Group Title

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 Group Title

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

5. experimentX Group Title

x_2, y_2 are the coordinates of C

6. experimentX Group Title

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

7. experimentX Group Title

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 Group Title

Algebra matters.

9. Callisto Group Title

C = (0, 2)

10. experimentX Group Title

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

11. Callisto Group Title

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

12. Callisto Group Title

he?

13. Callisto Group Title

Do you mean I knew the answer?

14. Callisto Group Title

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 Group Title

co-ordinates of A can also be found similarly.

16. Callisto Group Title

A = (-6, 0)

17. Callisto Group Title

D = (4, 0)

18. hartnn Group Title

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

19. hartnn Group Title

solving equation of line and circle simultaneously.

20. hartnn Group Title

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

21. Callisto Group Title

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 Group Title

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

23. Callisto Group Title

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

24. hartnn Group Title

no, u are correct.

25. Callisto Group Title

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

26. hartnn Group Title

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

27. shubhamsrg Group Title

did anyone notice : |dw:1356789155290:dw|

28. shubhamsrg Group Title

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 Group Title

how do we know slope of OB?

30. hartnn Group Title

ohh..

31. hartnn Group Title

as we know slope of OC

32. shubhamsrg Group Title

yep..exactly..

33. hartnn Group Title

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

34. Callisto Group Title

Not so relevant. |dw:1356789503661:dw|

35. shubhamsrg Group Title

ohh yes!! COA are collinear !

36. hartnn Group Title

how does that make it any simpler ?

37. shubhamsrg Group Title

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

38. Callisto Group Title

''Not so relevant.'' :P

39. hartnn Group Title

:P ok ..

40. hartnn Group Title

u got 2 more easy methods ;)

41. Callisto Group Title

Yes! Yes! Thanks! Thanks!!