find the solution set for each system: \frac{\left(x-1\right)^{2}}{16}+\frac{\left(y+2\right)^{2}}{9}=1 and \left(x-1\right)^{2}+\left(y+2\right)^{2}=16

We are given the system of equations:

\[
\frac{(x - 1)^2}{16} + \frac{(y + 2)^2}{9} = 1 \tag{1}
\]

\[
(x - 1)^2 + (y + 2)^2 = 16 \tag{2}
\]

### Step 1: Recognizing the equations
- Equation (1) is the equation of an ellipse, centered at \( (1, -2) \), with semi-major axis \( 4 \) along the \( x \)-axis and semi-minor axis \( 3 \) along the \( y \)-axis.
- Equation (2) is the equation of a circle, centered at \( (1, -2) \), with radius \( 4 \).

### Step 2: Simplifying the problem
Both curves are centered at \( (1, -2) \). The circle has a radius of 4, and the ellipse's semi-major axis along the \( x \)-axis is also 4, which suggests that these two curves intersect at points on the boundary of the circle and ellipse.

### Step 3: Substituting into Equation (2)
Let’s substitute parametric values from the ellipse into the circle equation. Start by simplifying each equation or testing boundary points common to both curves.

Let’s begin by looking for potential points that satisfy both. Since both are centered at \( (1, -2) \) and share boundary properties, we hypothesize symmetric or boundary intersections and then check them in the other equation.

### Step 4: Solving algebraically or testing intersections
Let’s attempt a direct solve from the derived information.from sympy import symbols, Eq, solve

# Define variables
x, y = symbols('x y')

# Equation 1 (ellipse)
eq1 = Eq((x - 1)**2 / 16 + (y + 2)**2 / 9, 1)

# Equation 2 (circle)
eq2 = Eq((x - 1)**2 + (y + 2)**2, 16)

# Solve the system of equations
solution = solve([eq1, eq2], (x, y))
solutionThe solution set for the given system of equations is:

\[
(x, y) = (-3, -2) \quad \text{and} \quad (x, y) = (5, -2)
\]

These two points satisfy both the equation of the ellipse and the equation of the circle.

Would you like more details on the steps? Here are five related questions to consider:
1. What are the general equations for ellipses and circles in the coordinate plane?
2. How can we determine if a point lies inside, on, or outside an ellipse?
3. What happens geometrically when a circle and an ellipse intersect?
4. Can a system like this have no real solutions? If so, why?
5. How do we use parametric equations to solve conic systems?

**Tip:** When solving systems of equations involving conics, symmetry can simplify finding solutions.

Find the solution set for each system: \frac{(x-1)^2}{16} + \frac{(y+2)^2}{9} = 1 and (x-1)^2 + (y+2)^2 = 16

Find the solution set for a system involving an ellipse and a circle. This problem explores key concepts from analytic geometry, including conic sections and systems of equations. Suitable for Grades 11-12 students.

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