	Curve	Cartesian Form	Parametric Form
1	Parabola	$y^2=4ax$	$x=at^2, \quad y=2at$
2	Ellipse	$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$	$x=a\cos \theta, \quad y= b \sin \theta$
3	Hyperbola	$\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$	$x= a \sec \theta, \quad y= b \tan \theta$

Director Circle:

Equation of Tangents

	Curve	Cartesian Form	Parametric Form	Slope Form
1	Parabola	$yy_1=2a(x+x_1)$	$yt=x+at^2$	$y=mx+\dfrac{a}{m}$
2	Ellipse	$\dfrac{xx_1}{a^2}+\dfrac{yy_1}{b^2}=1$	$\dfrac{x}{a}\cos \theta +\dfrac{y}{b}\sin \theta =1$	$y=mx+ \sqrt{a^2m^2+b^2}$
3	Hyperbola	$\dfrac{xx_1}{a^2}-\dfrac{yy_1}{b^2}=1$	$\dfrac{x}{a}\sec \theta- \dfrac{y}{b}\tan \thta =1$	$y=mx+\sqrt{a^2m^2-b^2}$

Comparative Study of Standard Equations of Parabola, Ellipse and Hyperbola

		Parabola	Ellipse	Hyperbola
1	Standard Equation	$y^2=4ax$	$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$	$\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$
2	Eccentricity	$e=1$	$e < 1$	$e > 1$
3	Relation between $a^2$ and $b^2$	–	$b^2=a^2 (1-e^2)$	$b^2=a^2(e^2-1)$
4	Centre	–	$(0,0)$	$(0,0)$
5	Focii	$(a, 0)$	$(ae, 0),\quad (-ae,0)$	$(ae, 0) \quad, (-ae, 0)$
6	Distance between focii	–	$2ae$	$2ae$
7	Vertices	$(0,0 )$	$(a, 0), \quad (-a ,0)$ $(0,b)\quad (0, -b)$	$(a, 0), \quad (-a, 0)$ $(0,b)\quad (0, -b)$
8	Axis	$x$ -axis	Major axis= $2a$ (along $x$ -axis ) Minor Axis= $2b$ (along $y$ -axis)	Transverse axis $=2a$ (along $x$ -axis) Conjugate Axis $=2b$ (along $y$ -axis)
9	Symmetry	About $x$ -axis	About origin and both axes	About origin and both axes

10	Equation of Directrices	$x+a=0$	$x= \pm \dfrac{a}{e}$	$x=\pm \dfrac{a}{e}$
11	Distance between Directrices	–	$\dfrac{2a}{e}$	$\dfrac{2a}{e}$
12	Length of Latus Rectum	$4a$	$\dfrac{2b^2}{a}$	$\dfrac{2b^2}{a}$

Application of Hyperbola:

Long-range navigation (LORAN) is a radio navigation system developed during World War II. The system enables a pilot to guide aircraft by maintaining a constant difference between the aircraft’s distances from two fixed points; the master station and the slave station.

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