Conic sections are the curves obtained when a plane cuts a double-napped right circular cone: a circle, ellipse, parabola or hyperbola according to the angle of the cut, with degenerate cases (a point, a line, a pair of lines) when the plane passes through the vertex. A circle is the set of points at fixed distance (radius) from a centre, with equation \((x-h)^2+(y-k)^2=r^2\). A parabola is the set of points equidistant from a focus and a directrix; the standard form \(y^2=4ax\) has focus \((a,0)\), directrix \(x=-a\) and latus rectum \(4a\). An ellipse is the set of points whose distances from two foci have constant sum; \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\) with \(c^2=a^2-b^2\), eccentricity \(e=c/a<1\) and latus rectum \(\dfrac{2b^2}{a}\). A hyperbola is the set of points whose distances from two foci have constant difference; \(\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\) with \(c^2=a^2+b^2\), eccentricity \(e=c/a>1\) and latus rectum \(\dfrac{2b^2}{a}\). These curves model planetary orbits, reflectors, arches and suspension cables.
Sections of a coneCircle and its equationParabola, focus, directrix and latus rectumEllipse, foci, eccentricity and latus rectumHyperbola, foci, eccentricity and latus rectumStandard equations and applications
Key terms
Conic section
A curve formed by intersecting a plane with a double-napped right circular cone.
Circle
Set of points at a fixed distance \(r\) from a centre; \((x-h)^2+(y-k)^2=r^2\).
Parabola
Set of points equidistant from a focus and a directrix; standard \(y^2=4ax\).
Ellipse
Set of points whose distances from two foci have constant sum; \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\).
Hyperbola
Set of points whose distances from two foci have constant difference; \(\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\).
