Callisto
  • 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
Mathematics
schrodinger
  • schrodinger
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Callisto
  • Callisto
|dw:1356781451262:dw|
Callisto
  • Callisto
brb
experimentX
  • 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
  • Callisto
May I know what (x2, y2) is??
experimentX
  • experimentX
x_2, y_2 are the coordinates of C
experimentX
  • experimentX
probably you would end up with two sets of point, choose the the point where y>0
experimentX
  • 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
  • Callisto
Algebra matters.
Callisto
  • Callisto
C = (0, 2)
experimentX
  • experimentX
yes!! sorry i was careless ... did you find out answer?
Callisto
  • Callisto
\(x^2 + y^2 +6x - 2y =0\) ---(1) \(x^2+(y-2)^2 =20\) => \(x^2 + y^2 -4y+4 =20\) ---(2)
Callisto
  • Callisto
he?
Callisto
  • Callisto
Do you mean I knew the answer?
Callisto
  • 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
  • hartnn
co-ordinates of A can also be found similarly.
Callisto
  • Callisto
A = (-6, 0)
Callisto
  • Callisto
D = (4, 0)
hartnn
  • hartnn
then u know equation of line BD can't u use that to find co-ordinates of B ?
hartnn
  • hartnn
solving equation of line and circle simultaneously.
hartnn
  • hartnn
are u sure about D =(4,0)??
Callisto
  • 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
  • hartnn
oh, i somehow got 36 instead of 40, thats why i was stuck.....
Callisto
  • Callisto
Correction: o.O Did I make a mistake? :S
hartnn
  • hartnn
no, u are correct.
Callisto
  • Callisto
Okay, thanks again! I didn't thought of the line. I'm just too stupid :(
hartnn
  • hartnn
happens! i just did a silly mistake of 6*4 =20 :P
shubhamsrg
  • shubhamsrg
did anyone notice : |dw:1356789155290:dw|
shubhamsrg
  • 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
  • hartnn
how do we know slope of OB?
hartnn
  • hartnn
ohh..
hartnn
  • hartnn
as we know slope of OC
shubhamsrg
  • shubhamsrg
yep..exactly..
hartnn
  • hartnn
given slope of OB and O, we can find B, no need of any distance..
Callisto
  • Callisto
Not so relevant. |dw:1356789503661:dw|
shubhamsrg
  • shubhamsrg
ohh yes!! COA are collinear !
hartnn
  • hartnn
how does that make it any simpler ?
shubhamsrg
  • shubhamsrg
i dont know,,just a good fact to know! :D
Callisto
  • Callisto
''Not so relevant.'' :P
hartnn
  • hartnn
:P ok ..
hartnn
  • hartnn
u got 2 more easy methods ;)
Callisto
  • Callisto
Yes! Yes! Thanks! Thanks!!

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