FACT SHEET: CONIC SECTIONSThe Conic EquationAx2+Bxy+Cy2+Dx+Ey+F

The ParabolaCartesian Form: y2=4ax (wherea is aConstant )Parametric Form: x=2at ; y=a t 2

Eccentricity: ε=1

Conic Discriminant: B2−4 AC=0

The EllipseCartesian Form: x2

a2+ y

2

b2=1 (wherea ,b areConstant s )

Parametric Form: x=a cosθ ; y=b sinθ

Eccentricity: ε<1

Conic Discriminant: B2−4 AC<0

The CircleCartesian Form: (x−a)2+( y−b)2=r2

(where (a ,b )is the ˚Centre ,∧r the Radius)Parametric Form: x=a+r cosθ ; y=b+r sinθ

Eccentricity: ε=0

Conic Discriminant: B2−4 AC<0

The HyperbolaCartesian Form: x2

a2− y2

b2=1 (wherea ,b areConstant s )

Parametric Form: x=a sec θ ; y=b tanθ

Eccentricity: ε>1

Conic Discriminant: B2−4 AC>0

Conic DistancesGiven the following: P: A Point on the Conic Curve (x p , yp)F: Is the Focus (x1 , y1)N: Is the Point on the Directrix which gives the shortest distance to the point P ax+by+c=0

The first thing to note is that line PN is Perpendicular to the Directrix. The Eccentricity: Is the constant Ratio between PN, the shortest line connecting the Point P to the Directrix, and PF, the line connecting the Point P to the Focus.

ε=¿PF∨ ¿¿ PN∨¿¿

¿

The line PN can be calculated via the following: ¿ PN∨¿2=

¿ ax p+b y p+c∨¿2

a2+b2=(x−x p )2+( y− y p )2¿¿Where a ,b , c are constants that define the Directrix, and (x p , yp) is the Point P.

The line PF can be calculated via the following:|PF|=ε2¿ PN∨¿2¿

We can find the Conic Equation of the Conic Section by equating the two instances of ¿ PN∨¿2¿:¿a x p+b y p+c∨¿2

a2+b2=(x−x p )2+( y− y p )2 ¿