Callisto
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
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Callisto
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Callisto
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brb
experimentX
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solve these equations for \( x_1, y_1 \),
\[ x_1^2 + y_1^2 + 6x_1−2y_1=0,\\
x_2^2 + y_2^2 + 6x_2 - 2y_1 = 0, x_2=0, y_2 \neq 0 \\
(x_1-x_2)^2 + (y_1-y_2)^2 = 20\]
Callisto
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May I know what (x2, y2) is??
experimentX
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x_2, y_2 are the coordinates of C
experimentX
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probably you would end up with two sets of point, choose the the point where y>0
experimentX
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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
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Algebra matters.
Callisto
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C = (0, 2)
experimentX
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yes!! sorry i was careless ... did you find out answer?
Callisto
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\(x^2 + y^2 +6x - 2y =0\) ---(1)
\(x^2+(y-2)^2 =20\) => \(x^2 + y^2 -4y+4 =20\) ---(2)
Callisto
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he?
Callisto
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Do you mean I knew the answer?
Callisto
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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
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co-ordinates of A can also be found similarly.
Callisto
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A = (-6, 0)
Callisto
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D = (4, 0)
hartnn
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then u know equation of line BD
can't u use that to find co-ordinates of B ?
hartnn
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solving equation of line and circle simultaneously.
hartnn
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are u sure about D =(4,0)??
Callisto
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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
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oh, i somehow got 36 instead of 40, thats why i was stuck.....
Callisto
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Correction: o.O Did I make a mistake? :S
hartnn
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no, u are correct.
Callisto
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Okay, thanks again! I didn't thought of the line. I'm just too stupid :(
hartnn
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happens!
i just did a silly mistake of 6*4 =20 :P
shubhamsrg
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did anyone notice :
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shubhamsrg
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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
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how do we know slope of OB?
hartnn
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ohh..
hartnn
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as we know slope of OC
shubhamsrg
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yep..exactly..
hartnn
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given slope of OB and O, we can find B, no need of any distance..
Callisto
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Not so relevant.
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shubhamsrg
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ohh yes!!
COA are collinear !
hartnn
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how does that make it any simpler ?
shubhamsrg
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i dont know,,just a good fact to know! :D
Callisto
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''Not so relevant.'' :P
hartnn
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:P
ok ..
hartnn
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u got 2 more easy methods ;)
Callisto
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Yes! Yes! Thanks! Thanks!!