Conic Section Formulas

Conic sections are curves formed by intersecting a plane with a cone. Understanding their equations—like circles, ellipses, parabolas, and hyperbolas—helps us analyze shapes and their properties, connecting geometry with algebra in practical ways.

1. Circle equation: (x - h)² + (y - k)² = r²

   • Represents all points equidistant from a center point (h, k).
   • r is the radius, determining the size of the circle.
   • The center (h, k) shifts the circle's position in the Cartesian plane.

2. Ellipse equation (horizontal): (x - h)²/a² + (y - k)²/b² = 1

   • Describes an elongated shape with a major axis along the x-axis.
   • a is the semi-major axis length, while b is the semi-minor axis length.
   • The center (h, k) indicates the midpoint of the ellipse.

3. Ellipse equation (vertical): (x - h)²/b² + (y - k)²/a² = 1

   • Represents an elongated shape with a major axis along the y-axis.
   • a is the semi-major axis length, while b is the semi-minor axis length.
   • The center (h, k) indicates the midpoint of the ellipse.

4. Parabola equation (vertical): (x - h)² = 4p(y - k)

   • Describes a U-shaped curve that opens upwards or downwards.
   • p is the distance from the vertex to the focus and directrix.
   • The vertex is located at (h, k), which is the turning point of the parabola.

5. Parabola equation (horizontal): (y - k)² = 4p(x - h)

   • Represents a U-shaped curve that opens to the left or right.
   • p is the distance from the vertex to the focus and directrix.
   • The vertex is located at (h, k), which is the turning point of the parabola.

6. Hyperbola equation (horizontal): (x - h)²/a² - (y - k)²/b² = 1

   • Describes two separate curves that open left and right.
   • a is the distance from the center to the vertices along the x-axis.
   • The center (h, k) is the midpoint between the two branches.

7. Hyperbola equation (vertical): (y - k)²/a² - (x - h)²/b² = 1

   • Represents two separate curves that open upwards and downwards.
   • a is the distance from the center to the vertices along the y-axis.
   • The center (h, k) is the midpoint between the two branches.

8. Eccentricity formula: e = c/a

   • Defines the measure of how much a conic section deviates from being circular.
   • c is the distance from the center to the focus.
   • For ellipses, e < 1; for parabolas, e = 1; for hyperbolas, e > 1.

9. Directrix equations for parabolas

   • Each parabola has a directrix, a line used to define its shape.
   • The distance from any point on the parabola to the focus equals the distance to the directrix.
   • The directrix is located p units away from the vertex, opposite the focus.

10. Focus-directrix definition of conic sections

• Conic sections are defined as the set of points equidistant from a focus and a directrix.
• This definition applies to all conic sections: circles, ellipses, parabolas, and hyperbolas.
• The focus is a fixed point, while the directrix is a fixed line, guiding the shape of the conic.