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 \ - \ ellipse,}

{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.

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