# Focus-Directrix Property of Conic Sections

To get the equation of a conic in terms of ${x}$ and ${y}$, we use a property, known as the focus-directrix property.

Let there be a fixed line ${l}$ (The Directrix) and a fixed point ${S}$ (The Focus). Let ${P}$ be a point on the locus of conic and let ${PM}$ be perpendicular to ${l}$. Clearly, as ${P}$ moves, ${M}$ slides on ${l}$. Now, consider the ratio

${\frac {SP}{PM} = e}$

If we do not allow this ratio to change, 4 possibilities arise, giving us distinct shapes:

${e=0 \ - \ circle, \ 0

${e=1 \ - \ parabola, \ e>1 \ - \ hyperbola}$

Axis of a conic is the line about which a conic is symmetric. A conic may have 1 or 2 or more axes of symmetry.

Vertex of a conic is the point of intersection of the conic with the axis.

Chord is any line segment, which joins 2 points on a curve. If the chord passes through the focus, it is known as focal chord.

Focal length is the length of segment which joins a point on the conic to the focus.

Latus rectum is the focal chord which is perpendicular to the axis.

Center of a conic is the point which bisects every chord of the conic passing through it.

Double ordinate of a conic is any chord, which is perpendicular to the axis.