anonymous
  • anonymous
How do u solve {x^2+y^2=63 x^2-3y^2=27
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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yuki
  • yuki
there are a couple of ways to solve this, but the most efficient way is to roughly see how the graph is and find out how many intersections there are going to be
yuki
  • yuki
the first eqn. is a circle with radius sqrt(63) and the second eqn. is a hyperbola
yuki
  • yuki
as you can easily see , both of the conics have the center located at the origin, so the number of intersection points can vary from none to 4

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yuki
  • yuki
let's put the hyperbola in the standard form \[{x^2 \over 27} - {y^2 \over 9} =1\]
yuki
  • yuki
from here we can observe that this hyperbola opens sideways, and the verteces are located sqrt(27) units right and left, 3 units up and down
yuki
  • yuki
oops, my bad, only sqrt(27) units right to left. the 3 units up and down tells us the asymptotes so never mind about that
yuki
  • yuki
since the verteces are located sqrt(27) units from the origin, and the circle has a radius of sqrt(63) we can see that the hyperbola crosses the circle four times
yuki
  • yuki
knowing this is actually important, because you don't want to waste time solving the system of eqn. only to find out there were no solutions.
yuki
  • yuki
anyway, now we will start solving this algebraically
yuki
  • yuki
the circle eqn can give us\[x^2 = 63-y^2\]
yuki
  • yuki
so if I substitute this into the hyperbola eqn \[(63 - y^2) - 3y^2 = 27\]
yuki
  • yuki
as you can see, everything in the eqn is in terms of y, so we can solve it without any trouble
yuki
  • yuki
since \[-4y^2 = -44 \] \[y = \pm 4\]
yuki
  • yuki
if you plug this into the circle, \[x^2 + (\pm 4)^2 = 63\] implies that \[x = \pm \sqrt{47}\]
yuki
  • yuki
unless my calculations are wrong, the four points that the circle and the hyperbola intersect are \[(\pm \sqrt{47},\pm 4)\]
yuki
  • yuki
wow, I did all this and skull is not even online :( oh well, let me know if anyone found this useful :)
anonymous
  • anonymous
\[\left\{x\to -3 \sqrt{6},y\to -3\right\},\left\{x\to -3 \sqrt{6},y\to 3\right\},\left\{x\to 3 \sqrt{6},y\to -3\right\},\left\{x\to 3 \sqrt{6},y\to 3\right\} \] A plot is attached. Used the following Mathematica statement to solve it \[\text{Solve}\left[\left\{x^2+y^2==63,x^2-3 y^2==27\right\},\{x,y\}\right] \] Plot expression follows: Plot[%87, {x, -10, 10}, AspectRatio -> Automatic] where %87 = \[\left\{-\sqrt{63-x^2},\sqrt{63-x^2},-\frac{\sqrt{-27+x^2}}{\sqrt{3}},\frac{\sqrt{-27+x^2}}{\sqrt{3}}\right\} \] yuki, Excellent advise with regard to having a plot to look at the solutions.
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