PRECAL - Conic Sections and Circles

Conic Section

The intersection of a plane in a double-napped cone

Parabola

Formed when the plane is parallel to the generating line of one cone

Circle

Formed when the plane is perpendicular to the axis of revolution

Ellipse

Formed when the plane intersects the one cone at an angle other than 90°

Hyperbola

Formed when the plane is parallel to the axis of revolution or the y-axis

Degenerate Conic Sections

Formed when a plane passes through the apex

Dot, Line, Two Intersecting Lines

Different types of Degenerate Conic Sections

Non-Degenerate Conic Sections

Formed when the plane does not pass through the apex

Parabola, Circle, Ellipse, Hyperbola

Different types of Non-Degenerate Conic Sections

Vertex

Extreme point of a parabola, hyperbola, and ellipse

Focus

Fixed point of a Parabola, Hyperbola and Ellipse

Directrix

Fixed Line of a parabola, hyperbola, and ellipse

x = -c

Formula to find the directrix of a parabola

x = a²/c

Formula to find the directrix of an ellipse and hyperbola

x² + y² = r²

Standard Formula of a Circle if the center is at the origin

(x-h)²+(y-k)²=r²

Standard Formula of a Circle if the center is NOT at the origin

Ax²+By²+Cx+Dy+E=0

General Formula of a Circle