Let S ( x1 , y1 ) be the focus, l the directrix, and e be the eccentricity. Let P ( x, y ) be the moving point.

**The general equation of a Conic**

Let *S *( *x*_{1} , *y*_{1} ) be
the focus, *l *the directrix, and *e *be the eccentricity. Let *P *(
*x*, *y *) be the moving point.

By the definition of conic, we have

and *PM *= perpendicular distance from *P*(*x*, *y*) to the line *lx *+ *my *+ *n *=
0

On simplification the above equation takes the form of general
second-degree equation

*Ax*^{2} + *Bxy *+ *Cy*^{2} + *Dx *+ *Ey *+ *F
*= 0 , where

yielding the following cases:

(i) *B*^{2} - 4 *AC *= 0 Û *e *= 1 Û the conic is a parabola,

(ii) *B*^{2} - 4 *AC *< 0 Û 0 < *e *<1 Û the conic is an ellipse,

(iii) *B*^{2} - 4 *AC *> 0 Û *e *>1 Û the conic is a hyperbola.

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12th Mathematics : UNIT 5 : Two Dimensional Analytical Geometry II : The general equation of a Conic |

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