## Real Solutions (x,y)

How  many real solutions $(x,y)$ are there that satisfy the two equations $x^2+y^2=30$ and $4y^2-x^2=100$?
Source: NCTM Mathematics Teacher, September 2006

Solution

The graph of $x^2+y^2=30$ is a circle centered at the origin and radius = $\sqrt{30}$. The graph of $4y^2-x^2=100$ is a hyperbola with vertices at $(0,5)$ and $(0,-5)$. The two graphs intersect at four points. There are four real solutions $(x,y)$ that satisfy the two equations.

Answer: $4$

Alternate solution
$x^2+y^2=30$
$4y^2-x^2=100$
————————
$5y^2=130$
$y^2=26$
$y=\pm\sqrt{26}$
Substitute the value of $y^2$ into the first equation
$x^2+26=30$
$x^2=4$
$x=\pm 2$