Graphed in the same standard (x,y) coodinate plane are a circle and a parabola. The circle has radius 3 and centre (0,0). The parabola has vertex (-3,-2), has a vertical axis of symmetry, and passes through (-2,-1). The circle and the parabola intersect at how many points?

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem of finding the number of intersection points between the given circle and parabola, we can follow these steps:

### Step 1: Write the equations of the circle and the parabola.

**Circle:**
The equation of a circle with center at (0,0) and radius 3 is given by:
\[ x^2 + y^2 = 3^2 \implies x^2 + y^2 = 9 \]

**Parabola:**
The parabola has a vertex at (-3, -2) and passes through the point (-2, -1). Since it has a vertical axis of symmetry, its equation can be written in the form:
\[ y = a(x + 3)^2 - 2 \]
To find the value of \(a\), we can use the point (-2, -1).

### Step 2: Substitute the point (-2, -1) into the parabola's equation.

Substituting \(x = -2\) and \(y = -1\):
\[ -1 = a(-2 + 3)^2 - 2 \]
\[ -1 = a(1)^2 - 2 \]
\[ -1 = a - 2 \]
\[ a = 1 \]

Thus, the equation of the parabola is:
\[ y = (x + 3)^2 - 2 \]

### Step 3: Set the equations equal to find intersection points.

Now, we need to find the points where the circle and the parabola intersect. We can substitute the expression for \(y\) from the parabola into the circle's equation:
\[ x^2 + ((x + 3)^2 - 2)^2 = 9 \]

### Step 4: Simplify and solve the equation.

First, expand the parabola's equation:
\[ y = (x + 3)^2 - 2 \implies y = x^2 + 6x + 9 - 2 \implies y = x^2 + 6x + 7 \]

Now substitute this into the circle's equation:
\[ x^2 + (x^2 + 6x + 7)^2 = 9 \]

Next, expand \((x^2 + 6x + 7)^2\):
\[ (x^2 + 6x + 7)^2 = x^4 + 12x^3 + 86x^2 + 84x + 49 \]
Thus, the equation becomes:
\[ x^2 + x^4 + 12x^3 + 86x^2 + 84x + 49 = 9 \]
\[ x^4 + 12x^3 + 87x^2 + 84x + 40 = 0 \]

### Step 5: Determine the number of real roots.

To find the number of intersection points, we need to determine the number of real roots of the polynomial \(x^4 + 12x^3 + 87x^2 + 84x + 40 = 0\).

Using Descartes' rule of signs or numerical methods (like graphing or using the Rational Root Theorem), we can analyze the polynomial.

### Conclusion:

After analyzing the polynomial, we find that it has two real roots, which indicates that the circle and the parabola intersect at **two points**.