10.7.1: Key Terms

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Key Terms

angle of rotation: an acute angle formed by a set of axes rotated from the Cartesian plane where, if $\cot (2 θ) > 0,$ then $θ$ is between $(0°, 45°);$ if $\cot (2 θ) < 0,$ then $θ$ is between $(45°, 90°);$ and if $\cot (2 θ) = 0,$ then $θ = 45°$

center of a hyperbola: the midpoint of both the transverse and conjugate axes of a hyperbola

center of an ellipse: the midpoint of both the major and minor axes

conic section: any shape resulting from the intersection of a right circular cone with a plane

conjugate axis: the axis of a hyperbola that is perpendicular to the transverse axis and has the co-vertices as its endpoints

degenerate conic sections: any of the possible shapes formed when a plane intersects a double cone through the apex. Types of degenerate conic sections include a point, a line, and intersecting lines.

directrix: a line perpendicular to the axis of symmetry of a parabola; a line such that the ratio of the distance between the points on the conic and the focus to the distance to the directrix is constant

eccentricity: the ratio of the distances from a point $P$ on the graph to the focus $F$ and to the directrix $D$ represented by $e = \frac{P F}{P D},$ where $e$ is a positive real number

ellipse: the set of all points $(x, y)$ in a plane such that the sum of their distances from two fixed points is a constant

foci: plural of focus

focus (of a parabola): a fixed point in the interior of a parabola that lies on the axis of symmetry

focus (of an ellipse): one of the two fixed points on the major axis of an ellipse such that the sum of the distances from these points to any point $(x, y)$ on the ellipse is a constant

hyperbola: the set of all points $(x, y)$ in a plane such that the difference of the distances between $(x, y)$ and the foci is a positive constant

latus rectum: the line segment that passes through the focus of a parabola parallel to the directrix, with endpoints on the parabola

major axis: the longer of the two axes of an ellipse

minor axis: the shorter of the two axes of an ellipse

nondegenerate conic section: a shape formed by the intersection of a plane with a double right cone such that the plane does not pass through the apex; nondegenerate conics include circles, ellipses, hyperbolas, and parabolas

parabola: the set of all points $(x, y)$ in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix

polar equation: an equation of a curve in polar coordinates $r$ and $θ$

transverse axis: the axis of a hyperbola that includes the foci and has the vertices as its endpoints

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