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I need to find the points of intersection of a circle with radius $2$ and centre at $(0,0)$ and a rectangular hyperbola with equation $xy=1$. As per the topic statement is there any way to solve this without the graphical method. I have tried setting the $y$ values equal but I cant solve the resulting equation for $x $.

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  • $\begingroup$ Just curious: what do you mean by the graphical method? How do you read out the value of the coordinates of the intersection points? $\endgroup$ – The Vee Nov 22 '16 at 12:24
  • $\begingroup$ By drawing respective graphs and then finding the points of intersections. $\endgroup$ – Batwayne Nov 27 '16 at 17:06
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The circle is described by

$$ x^2 + y^2 = 4 \tag{a} $$

and the hyperbola by

$$ y = 1/x \tag{b} $$

Replacing (b) into (a) you get

$$ x^2 + \frac{1}{x^2} = 4 \quad\Rightarrow\quad x^4 - 4x^2 + 1 = 0 $$

this is a quadratic equation in $x^2$ whose solutions are

$$ x^2 = 2 \pm \sqrt{3} $$

The intersection are then

$$ x = \pm(2 \pm 3^{1/2})^{1/2} \quad y = 1/x $$

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  • $\begingroup$ we will then get 4 points right? $\endgroup$ – Batwayne Nov 21 '16 at 17:11
  • $\begingroup$ @Batwayne Correct $x = \{2(2 + 3^{1/2})^{1/2}, 2(2 - 3^{1/2})^{1/2}, -2(2 + 3^{1/2})^{1/2}, -2(2 - 3^{1/2})^{1/2} \}$ and $y = 1/x$ $\endgroup$ – caverac Nov 21 '16 at 17:12
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You can also directly combine the equations into complete binomial formulas $$ (x+y)^2=x^2+y^2+2xy=4+2=6,\\ (x-y)^2=x^2+y^2-2xy=4-2=2 $$ and solve the trivial linear system for each of the 4 sign combinations of the roots.

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  • $\begingroup$ nice!$\quad\qquad$ $\endgroup$ – polfosol Nov 22 '16 at 5:30
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The equation of the circle is $x^2 + y^2 = 4$ and the equation of the hyperbola is $xy=1$

So the point of intersection would be a common solution to

$xy =1$

$x^2 + y^2 = 4$

so

$y = 1/x$

$x^2 + \frac 1{x^2} = 4$

$x^4 +1 = 4x^2$

$x^4 - 4x^2 + 1 = 0$

$x^2 = \frac {4 \pm \sqrt {12}}2$

$x^2 = 2 \pm \sqrt 3$

$x = \pm \sqrt{2 \pm \sqrt{3}}$

$y = 1/x = \pm \frac 1{\sqrt{2 \pm \sqrt{3}}}$

$= \pm \frac 1{\sqrt{2 \pm \sqrt{3}}}\frac {\sqrt {2\mp \sqrt {3}}}{\sqrt {2\mp\sqrt{3}}} $

$=\pm \frac{\sqrt {2\mp \sqrt {3}}}{\sqrt {4-3}}=\pm {\sqrt {2\mp \sqrt {3}}}$

So there are four points: $(\sqrt{2 + \sqrt{3}},{\sqrt{2 - \sqrt{3}}});(\sqrt{2 - \sqrt{3}},{\sqrt{2 +\sqrt{3}}});(-\sqrt{2 + \sqrt{3}},-{\sqrt{2 - \sqrt{3}}});(-\sqrt{2 - \sqrt{3}},-{\sqrt{2 + \sqrt{3}}});$

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  • $\begingroup$ You could simplify the $y$ coordinates using $1/(2\pm\sqrt{3}) = 2\mp\sqrt{3}$, which also shows the symmetry more clearly. $\endgroup$ – Klaus Draeger Nov 21 '16 at 21:22
  • $\begingroup$ I could. But I personally don't like that. It treats the pm sign as a distinct variable rather than a parity sign. Just my opinion. Anyway, that's why I listed the four points, distinctly. $\endgroup$ – fleablood Nov 21 '16 at 21:32
  • $\begingroup$ I actually meant writing the points as $(\sqrt{2+\sqrt{3}},\sqrt{2-\sqrt{3}})$ etc $\endgroup$ – Klaus Draeger Nov 21 '16 at 21:36
  • $\begingroup$ Oh, you're right. I maybe should have. $\endgroup$ – fleablood Nov 21 '16 at 21:39
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By trigonometry:

Any point on the circle has coordinates $(2\cos t,2\sin t)$. Then plugging in the other equation

$$4\sin t\cos t=1,$$

$$\sin 2t=\frac12,$$

giving

$$t\in{\frac\pi{12},\frac{5\pi}{12},\frac{13\pi}{12},\frac{17\pi}{12}}.$$


By hyperbolic trigonometry:

Let $x=e^t,y=e^{-t}$ be a parameteric solution of the equation of the hyperbola. (There is another branch with opposite signs.)

Then by the equation of the cirle

$$x^2+y^2=e^{2t}+e^{-2t}=2\cosh2t=4$$ and $$t=\pm\frac12\text{arcosh }2=\pm\frac12\ln(2+\sqrt3)=\pm\ln\sqrt{2+\sqrt3}.$$

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Solve x^2+y^2=4 xy=1

we get 4 solutions for x,y in 1st and 3rd quadrant and symetric around 0,0

(sqrt(6)+sqrt(2))/2,(sqrt(6)-sqrt(2))/2

(sqrt(6)-sqrt(2))/2,(sqrt(6)+sqrt(2))/2

-(sqrt(6)+sqrt(2))/2,-(sqrt(6)-sqrt(2))/2

-(sqrt(6)-sqrt(2))/2,-(sqrt(6)+sqrt(2))/2

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The circle is ,
$x^2 + y^2 = 4\tag{1}$ The hyperbola is,
$xy=1\tag{2}$ Adding 2xy on both sides of equation 1,

$$x^2+y^2+2xy=4+2xy$$ $$(x+y)^2=4+2xy$$ $$(x+y)^2=6\tag{since xy=1,from (1)}$$ $$x+y=\pm\sqrt6\tag{3}$$ putting 2 into 3 ,
$$x+1/x=\pm\sqrt6$$ $$x^2\pm\sqrt6x+1=0$$ solving this quadratic first for +$\sqrt6$ and then for -$\sqrt6$ we get 4 solutions for x,
$$\frac{\sqrt6+\sqrt2}2 ,\frac{\sqrt6-\sqrt2}2$$ and $$\frac{-\sqrt6+\sqrt2}2,\frac{-\sqrt6-\sqrt2}2$$ the solutions for y will be inverse of each of the solutions for x

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By a change of variable, $$x^2+y^2=4,\\xy=1$$ can be rewritten

$$X+Y=4,\\XY=1$$ provided you keep in mind that $x$ and $y$ have the same sign.

Then this is a classical sum/product problem, solved by

$$(X-Y)^2=(X+Y)^2-4XY=12,$$ then

$$X,Y=\frac{4\pm\sqrt{12}}2=2\pm\sqrt3.$$

Finally there are four solutions in $x,y$,

$$x=\color{blue}\pm\sqrt{2\color{green}\pm\sqrt3},\\y=\color{blue}\pm\sqrt{2\color{green}\mp\sqrt3}$$


More directly, multiply by $x^2$ and

$$x^2+y^2=4\implies x^4+x^2y^2=x^4+1=4x^2,$$ which is a biquadratic equation.


If you want, you can unnest the radicals by

$$\sqrt{2\pm\sqrt3}=\sqrt{\frac{4\pm2\sqrt3}2}=\sqrt{\frac{(\sqrt3\pm1)^2}2}=\frac{\sqrt3\pm1}{\sqrt2}.$$

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A system of equations such as yours, namely \begin{cases} x^2+y^2=4 \\ xy=1 \end{cases} is symmetric, because it doesn't change when $x$ and $y$ are swapped.

Rewrite $x^2+y^2=4$ as $(x+y)^2-2xy=4$, so you get \begin{cases} (x+y)^2=6 \\ xy=1 \end{cases} that can be divided into $$ \begin{cases} x+y=\sqrt{6} \\ xy=1 \end{cases} \qquad\text{or}\qquad \begin{cases} x+y=-\sqrt{6} \\ xy=1 \end{cases} $$ Solving the former suffices, for any solution of the former system provides a solution of the latter by changing signs.

The problem is now to find two numbers we know the sum and product of, that is, the roots of the quadratic equation $$ z^2-\sqrt{6}\,z+1=0 $$ The roots are $$ z=\frac{\sqrt{6}-\sqrt{2}}{2} \qquad\text{or}\qquad z=\frac{\sqrt{6}+\sqrt{2}}{2} $$

Thus we get the four points $$ \left(\frac{\sqrt{6}-\sqrt{2}}{2},\frac{\sqrt{6}+\sqrt{2}}{2}\right) \quad \left(\frac{\sqrt{6}+\sqrt{2}}{2},\frac{\sqrt{6}-\sqrt{2}}{2}\right) \\ \left(-\frac{\sqrt{6}-\sqrt{2}}{2},-\frac{\sqrt{6}+\sqrt{2}}{2}\right) \quad \left(-\frac{\sqrt{6}+\sqrt{2}}{2},-\frac{\sqrt{6}-\sqrt{2}}{2}\right) $$

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