Essential Conic Sections Equations to Know for Algebra 2

Conic sections are curves formed by the intersection of a plane and a cone. Understanding their equations—like circles, ellipses, hyperbolas, and parabolas—helps us analyze their shapes and properties, which is essential in Algebra 2 and beyond.

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 coordinate plane.

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

   • Describes a set of points where the sum of distances to two foci is constant.
   • a and b represent the semi-major and semi-minor axes, respectively.
   • The orientation of the ellipse depends on the values of a and b.

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

   • Represents two separate curves (branches) that open left and right.
   • The distance between the foci is greater than the distance between the vertices.
   • a determines the distance from the center to the vertices along the x-axis.

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

   • Similar to the horizontal hyperbola but opens upwards and downwards.
   • The center is at (h, k), with vertices along the y-axis.
   • The relationship between a and b affects the shape and spread of the branches.

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

   • Describes a curve that opens upwards or downwards based on the sign of p.
   • The vertex is at (h, k), and p indicates the distance from the vertex to the focus.
   • The directrix is a line that helps define the parabola's shape.

6. Parabola equation (horizontal axis of symmetry): (y - k)² = 4p(x - h)

   • Represents a parabola that opens to the left or right.
   • The vertex remains at (h, k), with p determining the distance to the focus.
   • The orientation is determined by the sign of p.

7. General form of a conic section: Ax² + Bxy + Cy² + Dx + Ey + F = 0

   • A unified representation of all conic sections (circle, ellipse, hyperbola, parabola).
   • The coefficients A, B, and C determine the type of conic section.
   • Can be used to derive the standard forms of conic sections through completing the square.

8. Eccentricity formula: e = c/a (for ellipses and hyperbolas)

   • Measures the "roundness" of a conic section; e < 1 for ellipses, e > 1 for hyperbolas.
   • c is the distance from the center to the foci, while a is the distance to the vertices.
   • Eccentricity helps classify the conic section's shape and properties.

9. Directrix equation for parabola (vertical axis): x = h ± p

   • Provides the location of the directrix line(s) for parabolas with vertical symmetry.
   • The directrix is used in conjunction with the focus to define the parabola.
   • The sign of p indicates the direction the parabola opens.

10. Directrix equation for parabola (horizontal axis): y = k ± p

• Indicates the position of the directrix line(s) for parabolas with horizontal symmetry.
• Similar to the vertical case, it helps define the parabola's shape.
• The value of p determines the distance from the vertex to the directrix.