Q. 41 - Precalculus Enhanced with Graphing Utilities

Question

graph each equation of the system. Then solve the system to find the points of intersection.

$$\begin{gathered} x^2+2y^2=16 \\ 4x^2-y^2=24 \end{gathered}$$

Step-by-Step Solution

Answer

The point of intersection is  $\left(\frac{8}{3},\frac{2\sqrt{10}}{3}\right),\left(\frac{8}{3},-\frac{2\sqrt{10}}{3}\right),\left(-\frac{8}{3},\frac{2\sqrt{10}}{3}\right),\left(-\frac{8}{3},-\frac{2\sqrt{10}}{3}\right)$.

Step 1. Given information

$$\begin{gathered} x^2+2y^2=16 \\ 4x^2-y^2=24 \end{gathered}$$

Step 2. Graphing each equation

$x^2+2y^2=16$

$4x^2-y^2=24$

Step 3. Calculation of intersection point

$$\begin{gathered} Consider, \\ {x}^2+2y^2=16......\left(1\right) \\ 4x^2-y^2=24.......\left(2\right) \\ subtracting eq. 2\times 2 from eq.1 \\ we get \\ 9x^2=64 \\ x=\pm \frac{8}{3} \\ Substituting value of x in eq1 to get y \\ x=\frac{8}{3} \\ y=\pm \frac{2\sqrt{10}}{3} \\ x=-\frac{8}{3} \\ y=\pm \frac{2\sqrt{10}}{3} \end{gathered}$$