Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Standard Equations

  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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