## Conic Sections

**Definition: **The curves obtained by intersection of a plane and a double cone in different orientation are called conic section.

### Definitions of various important terms:

**Focus:**The fixed point is called the focus of the conic-section.

**Directrix:**The fixed straight line is called the directrix of the conic section.**Eccentricity:**The constant ratio is called the eccentricity of the conic section and is denoted by e.

**Axis:**The straight line passing through the focus and perpendicular to the directrix is called the axis of the conic section. A conic is always symmetric about its axis.**Vertex:**The points of intersection of the conic section and the axis are called vertices of conic section.**Centre:**The point which bisects every chord of the conic passing through it, is called the centre of conic.**Latus-rectum:**The latus-rectum of a conic is the chord passing through the focus and perpendicular to the axis.

**Double ordinate:**The double ordinate of a conic is a chord perpendicular to the axis.**Focal chord:**A chord passing through the focus of the conic is called a focal chord.**Focal distance:**The distance of any point on the conic from the focus is called the focal distance of the point.

### General equation of a conic section when its focus, directrix and eccentricity are given:

Let S(α, β) be the focus, Ax + By + C = 0 be the directrix and e be the eccentricity of a conic.

Let P(h, k) be any point on the conic. Let PM be the perpendicular from P, on the directrix. Then by definition,

SP = ePM ⇒ SP^{2} = e^{2}PM^{2}

### Recognisation of conics

The equation of conics is represented by the general equation of second degree ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0 ……(i)

and discriminant of above equation is represented by ∆, where ∆ = abc + 2fgh – af^{2} – bg^{2} – ch^{2}

**Case I:** When ∆ = 0.

In this case equation (i) represents the degenerate conic whose nature is given in the following table.

S. No. | Condition | Nature of conic |

1. | ∆ = 0 and ab – h^{2} = 0 | A pair of coincident straight lines |

2. | ∆ = 0 and ab – h^{2} < 0 | A pair of intersecting straight lines |

3. | ∆ = 0 and ab – h^{2} > 0 | A point |

**Case II:** When ∆ ≠ 0.

In this case equation (i) represents the non-degenerate conic whose nature is given in the following table.

S. No. | Condition | Nature of conic |

1. | ∆ ≠ 0, h = 0, a = b, e = 0 | A circle |

2. | ∆ ≠ 0, ab – h^{2} = 0, e = 1 | A parabola |

3. | ∆ ≠ 0, ab – h^{2} > 0, e < 0 | An ellipse |

4. | ∆ ≠ 0, ab – h^{2} < 0, e > 0 | A hyperbola |

5. | ∆ ≠ 0, ab – h^{2} < 0, a + b = 0, e = √2 | A rectangular hyperbola |