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

- Callisto

- schrodinger

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- Callisto

|dw:1356781451262:dw|

- Callisto

brb

- experimentX

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

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## More answers

- Callisto

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

- experimentX

x_2, y_2 are the coordinates of C

- experimentX

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

- experimentX

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

- Callisto

Algebra matters.

- Callisto

C = (0, 2)

- experimentX

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

- Callisto

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

- Callisto

he?

- Callisto

Do you mean I knew the answer?

- 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.

- hartnn

co-ordinates of A can also be found similarly.

- Callisto

A = (-6, 0)

- Callisto

D = (4, 0)

- hartnn

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

- hartnn

solving equation of line and circle simultaneously.

- hartnn

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

- 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)

- hartnn

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

- Callisto

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

- hartnn

no, u are correct.

- Callisto

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

- hartnn

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

- shubhamsrg

did anyone notice :
|dw:1356789155290:dw|

- 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)

- hartnn

how do we know slope of OB?

- hartnn

ohh..

- hartnn

as we know slope of OC

- shubhamsrg

yep..exactly..

- hartnn

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

- Callisto

Not so relevant.
|dw:1356789503661:dw|

- shubhamsrg

ohh yes!!
COA are collinear !

- hartnn

how does that make it any simpler ?

- shubhamsrg

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

- Callisto

''Not so relevant.'' :P

- hartnn

:P
ok ..

- hartnn

u got 2 more easy methods ;)

- Callisto

Yes! Yes! Thanks! Thanks!!

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