Conic Sections Formulas to Know for Analytic Geometry and Calculus

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 shapes and their properties in analytic geometry and calculus, revealing their unique characteristics.

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

   • Represents a circle with center at point (h, k) and radius r.
   • All points (x, y) on the circle are equidistant from the center.
   • If r = 0, the circle collapses to a single point at (h, k).

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

   • Represents an ellipse centered at (h, k) with semi-major axis a and semi-minor axis b.
   • The sum of the distances from any point on the ellipse to the two foci is constant.
   • If a = b, the ellipse is a circle.

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

   • Represents a hyperbola centered at (h, k) that opens left and right.
   • The difference of the distances from any point on the hyperbola to the two foci is constant.
   • Asymptotes can be found using the equations y = k ± (b/a)(x - h).

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

   • Represents a hyperbola centered at (h, k) that opens up and down.
   • Similar properties to the horizontal hyperbola, with the roles of x and y reversed.
   • Asymptotes are given by y = k ± (a/b)(x - h).

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

   • Represents a parabola that opens upwards if p > 0 and downwards if p < 0.
   • The vertex is at (h, k) and the focus is at (h, k + p).
   • The directrix is the line y = k - p.

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

   • Represents a parabola that opens to the right if p > 0 and to the left if p < 0.
   • The vertex is at (h, k) and the focus is at (h + p, k).
   • The directrix is the line x = h - p.

7. Eccentricity formula: e = c/a

   • Eccentricity (e) measures the "roundness" of a conic section.
   • For circles, e = 0; for ellipses, 0 < e < 1; for parabolas, e = 1; for hyperbolas, e > 1.
   • c is the distance from the center to the foci, and a is the distance from the center to the vertices.

8. Focus-directrix form of a conic: e = distance from point to focus / distance from point to directrix

   • Defines conic sections based on the ratio of distances to a focus and a directrix.
   • This definition applies to all conic sections, providing a geometric interpretation.
   • The value of e determines the type of conic section.

9. General conic section equation: Ax² + Bxy + Cy² + Dx + Ey + F = 0

   • Represents all conic sections in a single equation.
   • The coefficients A, B, and C determine the type of conic (circle, ellipse, hyperbola, or parabola).
   • Can be transformed into standard form through completing the square or rotation of axes.

10. Discriminant for identifying conic type: B² - 4AC

• Used to classify conic sections based on the values of A, B, and C.
• If B² - 4AC < 0, the conic is an ellipse (or circle if A = C and B = 0).
• If B² - 4AC = 0, the conic is a parabola.
• If B² - 4AC > 0, the conic is a hyperbola.