Conic section

A conic section is defined as a locus of a point in a plane which moves in such a way, that the ratio of its distances from a fixed point (called the focus of a conic) and from a fixed line (called directrix of a conic) is a constant (called eccentricity of conic).

i.e $\frac{PS}{PN} = e$

Equation of a Conic

$Let P\equiv(h,k), S\equiv(x_1,y_1) and L\equiv Ax + By + C = 0$

$\frac{PS}{PN} \Rightarrow PS = ePN$
$\Rightarrow \sqrt{(h-x_1)^2 + (k - y_1)^2} = e| \frac{Ah+By+C}{\sqrt{A^2 + B^2}}|$
$\Rightarrow (h-x_1)^2 + (k-y_1)^2 = e^2 \frac{(Ah + Bk + C)^2}{A^2 + B^2}$

Hence, locus of moving point P(h,k), i.e. equation of conic is given by:

$(A^2 + B^2)\{(x-x_1)^2 + (y-y_1)^2\}=e^2(Ax + By + C)^2$

The possible conic sections are:

• circle
• ellipse
• parabola
• hyperbola
• point (degenerate)
• two intersecting lines (degenerate)
• two parallel lines (degenerate)
• nothing (degenerate)