Solve the nonlinear system of equations.
-$$\left\{ \begin{aligned}
x ^ { 2 } + y ^ { 2 } & = 81 \
y & = \frac { 1 } { 9 } x ^ { 2 } - 9
\end{aligned} \right.$$
A) $( - 9,0 ) , ( 0,9 ) , ( 0 , - 9 )$
B) $( - 9,0 ) , ( 9,0 ) , ( 0 , - 9 )$
C) $( - 9,0 ) , ( 9,0 )$
D) $( - 9,0 ) , ( 9,0 ) , ( 0,9 )$

Question

Quizplus · Accepted Answer

To solve the system, substitute $y$ from the second equation into the first equation: $x^2 + \left(\frac{1}{9}x^2 - 9\right)^2 = 81$. Simplifying this directly is complex, so instead, use the given second equation to express $y$ in terms of $x^2$, and then substitute back into the first equation. This yields a simpler path to the solution. However, recognizing that the system describes a circle ($x^2 + y^2 = 81$) and a parabola ($y = \frac{1}{9}x^2 - 9$), we can solve for intersections. Substituting $y$ from the second equation into the first gives a quadratic in $x^2$, which can be solved to find $x$ values. The solutions $x = -9, 9$ correspond to $y = 0$ when substituted back into the second equation. The option $y = -9$ does not satisfy the second equation when $x = 0$, thus the correct solutions are $(-9, 0)$ and $(9, 0)$, making option B correct.

Solve the Nonlinear System of Equations {x2+y2=81y=19x2−9\left\{ \begin{aligned}x ^ { 2 } + y ^ { 2 } & = 81 \\y & = \frac { 1 } { 9 } x ^ { 2 } - 9\end{aligned} \right.⎩⎨⎧​x2+y2y​=81=91​x2−9​

Solve the Nonlinear System of Equations $\left\{ \begin{aligned}x ^ { 2 } + y ^ { 2 } & = 81 \\y & = \frac { 1 } { 9 } x ^ { 2 } - 9\end{aligned} \right.$