Conic Sections Review Flashcard

A conic section is a curve obtained by intersecting a cone with a plane. The four types of conic sections are circles, ellipses, parabolas, and hyperbolas.

The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.

Eccentricity is a measure of how much a conic section deviates from being circular. It is defined as the ratio of the distance from a point on the conic to a focus and the distance from that point to the nearest directrix.

The standard form of an ellipse's equation is (x - h)²/a² + (y - k)²/b² = 1, where (h, k) is the center, a is the semi-major axis, and b is the semi-minor axis.

To find the center of a circle from its equation in the form x² + y² + Dx + Ey + F = 0, rearrange it to (x - h)² + (y - k)² = r², where h = -D/2 and k = -E/2.

An ellipse has two foci, a major axis, a minor axis, and its eccentricity is less than 1. The distance between the foci is related to the lengths of the axes.

The distance from the center to a focus (c) and the distance from the center to a vertex (a) are related by the equation c² = a² - b², where b is the semi-minor axis.

The standard form of a parabola's equation is y = a(x - h)² + k or x = a(y - k)² + h, where (h, k) is the vertex.

To convert a general quadratic equation Ax² + Bxy + Cy² + Dx + Ey + F = 0 to standard form, complete the square for the x and y terms.

A hyperbola is a type of conic section formed by the intersection of a plane and both halves of a double cone. It consists of two separate curves called branches.