What is the difference between identifying a parabola, ellipse, hyperbola and a circle?

To differentiate between a parabola, ellipse, hyperbola, and a circle, let's first understand the general form of each conic section:

1. **Parabola**:
- A parabola is a set of points that are equidistant from a fixed point (focus) and a fixed line (directrix).
- Standard form of a parabola:
- \( y = ax^2 + bx + c \) (vertical)
- \( x = ay^2 + by + c \) (horizontal)

2. **Ellipse**:
- An ellipse is the set of points where the sum of the distances from two fixed points (foci) is constant.
- Standard form of an ellipse:
- \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) (horizontal ellipse)
- \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \) (vertical ellipse)

3. **Hyperbola**:
- A hyperbola is the set of points where the difference of distances from two fixed points (foci) is constant.
- Standard form of a hyperbola:
- \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) (horizontal hyperbola)
- \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \) (vertical hyperbola)

4. **Circle**:
- A circle is the set of points that are equidistant from a fixed point (center).
- Standard form of a circle:
- \( x^2 + y^2 = r^2 \)

### Differences:
- **Parabola**: Defined by one focus and one directrix, typically forms a U-shaped curve.
- **Ellipse**: Defined by two foci, forms an elongated oval shape.
- **Hyperbola**: Defined by two foci, forms two separate curves that open away from each other.
- **Circle**: Defined by a single center, forms a closed, symmetrical curve.

Each type of conic section has distinct properties and forms, based on the focus and directrix or foci used in the equation.