Questions: Choose the solution (intersection) of this quadratic system of inequalities.
x^2 + y^2 ≤ 9 and y ≥ -x^2 + 2
Transcript text: Choose the solution (intersection) of this quadratic system of inequalities.
\[
x^{2}+y^{2} \leq 9 \text { and } y \geq-x^{2}+2
\]
Solution
Solution Steps
Step 1: Analyze the first inequality
The first inequality, \(x^2 + y^2 \le 9\), represents the area inside and on a circle centered at the origin with a radius of 3.
Step 2: Analyze the second inequality
The second inequality, \(y \ge -x^2 + 2\), represents the area above and on the parabola \(y = -x^2 + 2\), which opens downwards and has a vertex at (0, 2).
Step 3: Find the intersection
The solution to the system of inequalities is the intersection of the regions defined by each inequality. This is the region that lies inside or on the circle and is also above or on the parabola. This corresponds to the second option.