Question 2 of 5 Which system of conic sections can have 2 solutions? Draw a picture of each to support you!

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In the study of conic sections, a **system of conic sections** refers to two different conic equations simultaneously. The solutions to this system are the points where the two conics intersect. Depending on their types and positions, two conic sections can intersect at zero, one, two, three, or four points. When a system has **exactly two solutions**, it means the two conic sections intersect at two distinct points.

Below are some common systems of conic sections that can have exactly two solutions, along with descriptive illustrations to help visualize each scenario.

---

### 1. **Circle and Circle**

**Description:**
When two circles overlap such that they cross each other in two distinct points, the system has two solutions.

**Visualization:**

Imagine two circles of different sizes overlapping like a Venn diagram. The overlapping area has two points where the circles intersect.

```
   ***       ***
 *     *   *     *
*       * *       *
*        *        *
 *             *
   *         *
     *     *
       ***
```

**Example Equations:**
- Circle 1: $$ x^2 + y^2 = 25 $$ (radius 5)
- Circle 2: $$ (x-8)^2 + y^2 = 9 $$ (radius 3, centered at (8, 0))

These circles intersect at two points.

---

### 2. **Ellipse and Line**

**Description:**
A straight line that cuts through an ellipse will typically intersect it at two points.

**Visualization:**

Picture an oval (ellipse) with a horizontal or vertical line passing through its center, crossing it at two points.

```
       *****
     **     **
    *         *
    *         *
     **     **
       *****

|
         |
```

**Example Equations:**
- Ellipse: $$ \frac{x^2}{16} + \frac{y^2}{9} = 1 $$
- Line: $$ y = 2 $$

The line intersects the ellipse at two points where $$ y = 2 $$.

---

### 3. **Parabola and Line**

**Description:**
A line that intersects a parabola (and is not parallel to its axis) will typically meet it at two points.

**Visualization:**

Imagine a "U-shaped" parabola with a straight line cutting through it from one side to the other, intersecting at two points.

```
        *
       * *
      *   *
     *     *
    *       *
   ***********
       |
       |
```

**Example Equations:**
- Parabola: $$ y = x^2 $$
- Line: $$ y = 2x + 3 $$

Solving these equations simultaneously yields two intersection points.

---

### 4. **Hyperbola and Circle**

**Description:**
A hyperbola can intersect a circle at two points if they are positioned such that one branch of the hyperbola crosses the circle twice.

**Visualization:**

Picture a hyperbola with two branches opening left and right, and a circle positioned so that one branch intersects the circle at two points.

```
   \       /
    \     /
     \   /
      \ /
       *
      / \
     /   \
    /     \
   /       \
```

**Example Equations:**
- Hyperbola: $$ \frac{x^2}{16} - \frac{y^2}{9} = 1 $$
- Circle: $$ (x-2)^2 + y^2 = 4 $$

These equations intersect at two points.

---

### 5. **Two Parabolas**

**Description:**
Two parabolas opening in opposite directions can intersect at two points.

**Visualization:**

Imagine one parabola opening upwards and another opening downwards, intersecting each other at two distinct points.

```
    *         *
   * *       * *
  *   *     *   *
 *     *   *     *
*       * *       *
         *
         *
```

**Example Equations:**
- Parabola 1: $$ y = x^2 $$ (opening upwards)
- Parabola 2: $$ y = -x^2 + 4 $$ (opening downwards)

These parabolas intersect at two points where their $$ y $$-values are equal.

---

### Summary

These systems demonstrate how different pairs of conic sections can intersect exactly twice:

1. **Circle & Circle**
2. **Ellipse & Line**
3. **Parabola & Line**
4. **Hyperbola & Circle**
5. **Two Parabolas**

Each system's specific positioning and orientation determine whether they will intersect at two points. Visualizing these scenarios can greatly aid in understanding their interactions.

---

*Note:* For more accurate and detailed visual representations, graphing tools or software (like Desmos or GeoGebra) can be used to plot the equations and observe their intersections.
Two conic sections can intersect at two points in several ways, such as two circles overlapping, a circle and a line intersecting, a parabola and a line intersecting, a hyperbola and a circle intersecting, or two parabolas intersecting each other.

Question 2 of 5 Which system of conic sections can have 2 solutions? Draw a picture of each to support you!

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