**MATHS :: Lecture 04 :: Conic sections**

**Conics**

**Definition**

A conic is defined as the locus of a point, which moves such that its distance from a fixed line to its distance from a fixed point is always constant. The fixed point is called the focus of the **conic**. The fixed line is called the **directrix **of the conic. The constant ratio is the **eccentricity** of the conic.

L is the fixed line – Directrix of the conic.

F is the fixed point – Focus of the conic.

constant ratio is called the eccentricity = ‘e’

**Classification of conics with respect to eccentricity**

1. If e < 1, then the conic is an Ellipse

** **

1) The ** standard equation** of an ellipse is

2) The line segment AA1 is the

**of the ellipse, AA1 = 2a**

*major axis*3) The equation of the major axis is Y = 0

4) The line segment BB1 is the

**of the ellipse, BB1 = 2b**

*minor axis*5) The equation of the minor axis is X = 0

6) The length of the major axis is always

**the minor axis.**

*greater than*7) The point O is the intersection of major and minor axis.

8) The co-ordinates of O are (0,0)

9) The

**of the ellipse are S(ae,0)and SI(-ae,0)**

*foci*10) The vertical lines passing through the focus are known as

*Latusrectum*11) The length of the Latusrectum is

12) The points A (a,0) and A1(-a,0)

13) The

**of the ellipse is e =**

*eccentricity*14) The vertical lines M1M11and M2M21 are known as the

**of the ellipse and their respective**

*directrix*equations are x = and x =

2. If e = 1, then the conic is a

**Parabola**

**.**

- The
of the parabola is y2 = 4ax.*Standard equation* - The horizontal line is the
.*axis of the parabola* - The equation of the axis of the parabola is Y = 0
- The parabola y2 = 4ax is
about the axis of the parabola.*symmetric* - The
of the parabola is O (0,0)*vertex* - The line PQ is called the
of the parabola.*directrix* - The equation of the directrix is x = -a
- The
of the parabola is S(a,0).*Focus* - The vertical line passing through S is the
. LL1 is the Latus rectum and its length LL1 = 4a*latus rectum*

3. If e > 1 ,then the conic is *Hyperbola*

*.*

1) The **standard equation** of an hyperbola is

2) The line segment AA1 is the **Transverse axis** of the hyperbola ,AA1 = 2a

3) The equation of the **Transverse axis** is Y = 0

4) The line segment BB1 is the **Conjugate axis** of the hyperbola ,BB1 = 2b

5) The equation of the **Conjugate axis** is X = 0

6) The point O is the intersection of **Transverse** and **Conjugate** axis.

7) The co-ordinates of O are (0,0)

8) The **foci **of the hyperbola are S(ae,0)and SI(-ae,0)

9) The vertical lines passing through the focus are known as **Latusrectum**

10) The length of the Latusrectum is

11) The points A (a,0) and A1(-a,0)

12) The **eccentricity** of the hyperbola is e =

13) The vertical lines M1M11and M2M21 are known as the **directrix** of the hyperbola and their respective equations are x = and x =

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