# Real Solutions

• Oct 26th 2006, 06:37 PM
Rimas
Real Solutions
How many real solutions [x,y] are there that satisfy the two equations x^2 + y^2= 30 and 4y^2-X^2=100?
• Oct 26th 2006, 06:44 PM
ThePerfectHacker
You have a circle and hyperbola.
• Oct 26th 2006, 10:30 PM
Soroban
Hello, Rimas!

Did you try solving the system?

Quote:

How many real solutions $(x,y)$ are there that satisfy: . $\begin{array}{cc}(1)\\(2)\end{array}\;\begin{array }{cc}x^2 + y^2\:=\:30 \\ 4y^2-x^2\:=\:100\end{array}$

Add the equations: . $5y^2 = 130\quad\Rightarrow\quad y^2 = 26\quad\Rightarrow\quad y = \pm\sqrt{26}$

Substitute into (1): . $x^2 + 26 \:=\:30\quad\Rightarrow\quad x^2 = 4\quad\Rightarrow\quad x = \pm2$

There are four solutions: . $(2,\,\sqrt{26}),\;(2,\,\text{-}\sqrt{26}),\;(\text{-}2,\,\sqrt{26}),\;(\text{-}2,\,\text{-}\sqrt{26})$