How many real solutions [x,y] are there that satisfy the two equations x^2 + y^2= 30 and 4y^2-X^2=100?
Hello, Rimas!
Did you try solving the system?
How many real solutions $\displaystyle (x,y)$ are there that satisfy: .$\displaystyle \begin{array}{cc}(1)\\(2)\end{array}\;\begin{array }{cc}x^2 + y^2\:=\:30 \\ 4y^2-x^2\:=\:100\end{array}$
Add the equations: .$\displaystyle 5y^2 = 130\quad\Rightarrow\quad y^2 = 26\quad\Rightarrow\quad y = \pm\sqrt{26}$
Substitute into (1): .$\displaystyle x^2 + 26 \:=\:30\quad\Rightarrow\quad x^2 = 4\quad\Rightarrow\quad x = \pm2$
There are four solutions: .$\displaystyle (2,\,\sqrt{26}),\;(2,\,\text{-}\sqrt{26}),\;(\text{-}2,\,\sqrt{26}),\;(\text{-}2,\,\text{-}\sqrt{26}) $