Key Terms

Key Terms

angle of rotation

an acute angle formed by a set of axes rotated from the Cartesian plane where, if $\cot \left(2\theta \right)>0,$ then $\theta$ is between $(\mathrm{0°},\mathrm{45°});$ if $\cot (2\theta )<0,$ then $\theta$ is between $(\mathrm{45°},\mathrm{90°});$ and if $\cot \left(2\theta \right)=0,$ then $\theta =\mathrm{45°}$

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 $e=\frac{PF}{PD},$ where $e$ is a positive real number

ellipse

the set of all points $\left(x,y\right)$ 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 $\left(x,y\right)$ on the ellipse is a constant

hyperbola

the set of all points $\left(x,y\right)$ in a plane such that the difference of the distances between $\left(x,y\right)$ 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 $\left(x,y\right)$ 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 $\theta$

transverse axis

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