Conic Sections

1. Definition:

The locus of a point P such that its distance from a fixed point S bears a constant ratio 'e' to its distance from a fixed line is called a conic section or conic.
The fixed point S is called the focus, the fixed line is called the directrix and the constant ratio 'e' is called the eccentricity of the conic.

The conic is called a parabola if e = 1

The conic is called an ellipse if e $<$ 1

The conic is called a hyperbola if e $>$ 1

Parabola
2. The equation of the parabola in standard form is $y^{2}$ = 4ax. ( a $>$ 0) (Right)

(a) Focus (a, 0)

(b) Focus (0,0)

(c) Directrix, x= -a

(d) Length of the latus rectum = the focal chord = 4a

3. Other forms of the parabola are:

$y^{2}$ = -4ax (a $>$ 0) (Left)

$x^{2}$ = 4ay (a $>$ 0) (Upward)

$x^{2}$ = -4ay (a $< >$ 0) (Downward)

4. Parametric co-ordinates of a point on the parabola $y^{2}$ = 4ax are ( ${at}^{2}$ , 2at) where t is any parameter

5. The equation of the chord joining P $({at}_{1}^{2}, {2 at}_{1})$ and Q $({at}_{2}^{2}, {2 at}_{2})$ is y( $t_{1} + t_{2}$ ) = 2x + ${2 at}_{1}$ $t_{2}$ .

For PQ to be focal chord $t_{1} t_{2}$ = -1 and the length of the focal chord having $t_{1}$ and $t_{2}$

as end points is a ${(t_{2} - t_{1})}^{2}$ .

6. Tangent

${yy}_{1}$ = 2a (x + $x_{1}$ ) at ( $x_{1}$ , $y_{1}$ )

yt= x + ${at}^{2}$ at ( ${at}^{2}$ , 2at)

y= mx + $\frac{a}{m}$ at $(\frac{a}{m^{2}}, \frac{2 a}{m})$

The st. line lx + my + n= 0 touches $y^{2}$ = 4ax if nl= ${am}^{2}$ , x cos $α$ + y sin $α$ = p touches

if p cos $α$ + a $\sin^{2}$ $α$ = 0.

7. Point of intersection of tangents at ' $t_{1}$ ' and ' $t_{2}$ ' is $[a t_{1} t_{2}, a (t_{1} + t_{2})]$

8. Normal

y- $y_{1}$ = $\frac{{- y}_{1}}{2 a}$ (x - $x_{1}$ ) at ( $x_{1}$ , $y_{1}$ )

y + tx= 2at + ${at}^{3}$ at 't'

y= mx - 2am - ${am}^{3}$ at ( ${am}^{2}$ , -2am)

9. Point of intersection of two normals at ' $t_{1}$ ' and ' $t_{2}$ ' is $[2 a + a (t_{1}^{2} + t_{2}^{2} + t_{1} t_{2}), a t_{1} t_{2} (t_{1} + t_{2})]$

10. Condition that the line y = mx + c to be a tangent or normal to the parabola $y^{2}$ = 4ax is c= a/m for tangent c= -2am - a $m^{3}$ for normal.

11. Chord with a given middle point ( $x_{1}$ , $y_{1}$ ) is $T_{1}$ = $S_{1}$ ie., ${yy}_{1}$ - 2a (x + $x_{1}$ ) = $y_{1}^{2}$ - 4 ${ax}_{1}$

12. The locus of the midpoint of a system of parallel chords of a parabola is called its diameter, its eqn. is y = 2a/m, if slope of the chord is 'm'.

13. Tangent at the extremities of any focal chord of a parabola meet at right angles on the directrix. Normals at the end points of latus rectum of the parabola meet at right angles on the axis of the parabola.

Ellipse
A. The equation of the ellipse in standard form is $\frac{x^{2}}{a^{2}}$ + $\frac{y^{2}}{b^{2}}$ = 1

The lines ${AA}^{/}$ and ${BB}^{/}$ are called major and minor axes respectively. A and $A^{/}$ are

called vertices of the ellipse. C is called the centre of the ellipse.

Form $\frac{x^{2}}{a^{2}}$ + $\frac{y^{2}}{b^{2}}$ = 1( $a^{2}$ $>$ $b^{2}$ ) $\frac{x^{2}}{b^{2}} + \frac{y^{2}}{a^{2}}$ = 1( $a^{2}$ $>$ $b^{2}$ )

1. Centre (0,0) (0,0)

2. Foci ( $\pm$ ae, 0) (0, $\pm$ ae)

3. Length of the major axis 2a 2a

4. Length of the minor axis 2b 2b

5. Equation of the major axis y = 0 x = 0

6. Equation of the minor axis x = 0 y = 0

7. Equation of the directrices x= $\pm$ $\frac{a}{e}$ y= $\pm$ $\frac{a}{e}$

8. Length of lactus rectum $\frac{{2 b}^{2}}{a}$ $\frac{{2 b}^{2}}{a}$

9. Equation of the lactus rectum x= $\pm$ ae y= $\pm$ ae

10. Vertices ( $\pm$ a, 0) ( $\pm$ b, 0)

11. Ends of minor axis (0, $\pm$ b) ( $\pm$ b, 0)

12. The focal distance of any point (x,y) is a - ex, a+ex a - ey, a+ey

13. $b^{2}$ = $a^{2}$ (l - $e^{2}$ ) or $e^{2}$ = 1 - $b^{2}$ / $a^{2}$ , (e $<$ 1)

B. Condition of tangency; y = mx + c is tangent if $c^{2}$ = $a^{2} m^{2}$ + $b^{2}$

C. Tangent in terms of the slope 'm' are y= mx $\pm$ $\sqrt{a^{2} m^{2} + b^{2}}$

D. Focal distance a + ex, a -ex; sum of focal distance = 2a = length of major axis.

E. Th equation of the tangent to the ellipse at ( $x_{1}$ , $y_{1}$ ) and the eqn. of the chord of contact of tangents drawn from an external point ( $x_{1}$ , $y_{1}$ ) is $T_{1}$ = $\frac{{XX}_{1}}{a^{2}}$ + $\frac{{YY}_{1}}{b^{2}}$ = 1

F. The point (a cos $θ$ , b sin $θ$ ) to the ellipse is $\frac{x}{a}$ cos $θ$ + $\frac{y}{b}$ sin $θ$ = 1

H. The normal at (a cos $θ$ , b sin $θ$ ) to the ellipse is $\frac{ax}{\cos θ}$ - $\frac{by}{\sin θ}$ = $a^{2} - b^{2}$

I. Auxiliary circle. $x^{2}$ + $y^{2}$ = $a^{2}$ .It is a circle with major axis as diameter.

J. Director circle. $x^{2}$ + $y^{2}$ = $a^{2}$ + $b^{2}$ . It is the locus of the point of intersection of two perpendicular tangents to an ellipse.

K. The chord with a given mid-point ( $x_{1}$ , $y_{1}$ ) is $\frac{{XX}_{1}}{a^{2}}$ + $\frac{{YY}_{1}}{b^{2}}$ = $\frac{x_{1}^{2}}{a^{2}}$ + $\frac{y_{1}^{2}}{b^{2}}$ ie., $T_{1}$ = $S_{1}$

L. Tangent at the extremities of latus rectum of an ellipse intersect on the corresponding directrix.

M. Product of the perpendicular from the foci upon any tangent to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}}$ =1 is constant = $b^{2}$ .

N. Four normals can be drawn from a point P to an ellipse and called co-normal points.

O. The sum of the eccentric angles of the co-normal points on the ellipse $\frac{x^{2}}{a^{2}}$ + $\frac{y^{2}}{b^{2}}$ =1 is equal to an odd multiple of $π$ ie (2n + 1) $π$ .

P. If $α$ , $β$ , $γ$ are eccentric angles of three points on the ellipse the normals at which are concurrent then sin ( $α$ + $β$ ) + sin ( $β$ + $γ$ ) + sin ( $γ$ + $α$ )= 0

Q. In an ellipse there are two foci S (ae, 0) and S' (ae, 0). If P is any point on the ellipse, then PS + PS' = 2a, length of the major axis.

Hyperbola
Equation of the hyperbola in standard form is $\frac{x^{2}}{a^{2}}$ - $\frac{y^{2}}{b^{2}}$ = 1. Transverse axis along x - axis and conjugate axis along y-axis.

Variuos results and properties
1. Vertices A(a, 0) $A^{/}$ (-a, 0) B(0, b) $B^{/}$ (0, -b)
2. Eccentricity $e^{2}$ = 1+ $\frac{b^{2}}{a^{2}}$ .
3. Equation of the directions are x= $\pm$ a/e and asymptotes are y = $\pm$ (b/e) . x
4. Length of latus rectum = $\frac{{2 b}^{2}}{a}$ , and its eqn. is x = $\pm$ ae
5. Focal distance: ex - a, ex + a
6. Difference of focal distance is equal to Length of transverse axis, ie, S'P - SP= 2a where P is any point on hyperbola
7. Parametric equation x = a sec $θ$ ; y= b tan $θ$ .
8. Equation of the tangent at ( $x_{1}$ , $y_{1}$ ) is $\frac{{XX}_{1}}{a^{2}}$ - $\frac{{YY}_{1}}{b^{2}}$ = 1
9. Equation of the tangent at ( a sec $θ$ , b tan $θ$ ) is $\frac{x}{a}$ sec $θ$ - $\frac{y}{b}$ tan $θ$ = 1and normal is ax cos $θ$ + by cot $θ$ = $a^{2}$ + $b^{2}$ .
10. Condition that the line y= mx + c be a tangent to the ellipse is $c^{2}$ = $a^{2}$ $m^{2}$ - $b^{2}$ . Equation of the tangent in terms of the slope m is y = mx $\pm$ $\sqrt{a^{2} m^{2} - b^{2}}$
11. Auxiliary circle $x^{2}$ + $y^{2}$ = $a^{2}$
12. Director circle $x^{2}$ + $y^{2}$ = $a^{2}$ - $b^{2}$
13. Chord with a given point ( $x_{1}$ , $y_{1}$ ) as middle point is $T_{1}$ = $S_{1}$ , $\frac{{XX}_{1}}{a^{2}}$ - $\frac{{YY}_{1}}{b^{2}}$ = $\frac{X_{1}^{2}}{a^{2}}$ - $\frac{Y_{1}^{2}}{b^{2}}$
14. Two tangents can be drawn from a point to a hyperbolam and four normals from a point to a hyperbola.

15. The line lx + my + n= 0 touches the hyperbola $\frac{x^{2}}{a^{2}}$ - $\frac{y^{2}}{b^{2}}$ = 1 if $a^{2 | 2}$ - $b^{2}$ $m^{2}$ = $n^{2}$
16. Centre of the hyperbola = midpoint of the line joining two foci.
17. Equation of the transverse axis is y= 0 and its length = 2a.
18. Length of the conjugate axis is 2b and its eqn. is x = 0.sssss

1.	Centre	(0,0)	(0,0)
2.	Foci	( $\pm$ ae, 0)	(0, $\pm$ ae)
3.	Length of the major axis	2a	2a
4.	Length of the minor axis	2b	2b
5.	Equation of the major axis	y = 0	x = 0
6.	Equation of the minor axis	x = 0	y = 0
7.	Equation of the directrices	x= $\pm$ $\frac{a}{e}$	y= $\pm$ $\frac{a}{e}$
8.	Length of lactus rectum	$\frac{{2 b}^{2}}{a}$	$\frac{{2 b}^{2}}{a}$
9.	Equation of the lactus rectum	x= $\pm$ ae	y= $\pm$ ae
10.	Vertices	( $\pm$ a, 0)	( $\pm$ b, 0)
11.	Ends of minor axis	(0, $\pm$ b)	( $\pm$ b, 0)
12.	The focal distance of any point (x,y) is	a - ex, a+ex	a - ey, a+ey