# Conic Sections Class 11 Notes Mathematics Chapter 11 - CBSE

## Chapter : 11

## What Are Conic Sections ?

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## Circle

A circle is a set of all points in a plane that are equidistance from a fixed point in the plane.

The equation of any circle whose centre and radius are given is

(x – h)^{2} + (y – k)^{2} = a^{2 }

## General Equation Of A Circle

The general equation of a circle is

x^{2} + y^{2} + 2gx + 2fy + c = 0

whose centre is (−g,−f) and radius

$$=\sqrt{g^{2} + f^{2} - c}$$

## Diameter Form Of A Circle

The equation of circle drawn on the straight line joining two given points (x_{1} , y_{1}) and (x_{2} , y_{2}) as diameter is (x – x_{1}) (x – x_{2}) + (y – y_{1}) (y – y_{2}) = 0

## Parabola

Parabola is a locus of a point which is equidistant from a fixed point (called focus) and a fixed line (called directrix). Thus, if (α, β) is the focus and ax + by + c = 0 is the equation of the directrix of a parabola, then the equation is

$$(x-α)^{2} + (x-β)^{2}\\=\frac{(ax + by + c)^{2}}{a^{2} + b^{2}}$$

This equation is of the form

ax^{2} + 2hxy + by^{2}+ 2gx + 2fy + c = 0,

satisfying the condition

abc + 2fgh – af^{2} – bg^{2} – ch^{2} ≠ 0 and h^{2} = ab. Standard form of a parabola is

y^{2} = 4ax

## Latus Rectum

Double ordinate through the focus is called the latus-rectum i.e. latusrectum is a chord passing through the focus perpendicular to the axis.

## Focal Distance

The distance P (x, y) from the focus S is called the focal distance of the point P.

∴ SP = a + x.

**Some other Standard Form of Parabola**

y^{2} = 4ax | y^{2} = -4ax | x^{2} = 4ay | x^{2} = 4ay | |

Coordinate of vertex | (0, 0) | (0, 0) | (0, 0) | (0, 0) |

Coordinate of focus | (a, 0) | (−a, 0) | (0, a) | (0, −a) |

Equation of directrix | x = −a | x = a | y = −a | y = a |

Equation of the axis | y = 0 | y = 0 | x = 0 | x = 0 |

Length of the latus-rectum | 4a | 4a | 4a | 4a |

Focal distance of a point P(x, y) | a + x | a − x | a + y | a − y |

## An Ellipse

An ellipse is the locus of a point in a plane which moves in such a way that the ratio of its distance from a fixed point (called focus) in the same plane to its distance from a straight line (called directrix) is always less than unity. The constant ratio of generally denoted by e and is known as eccentricity of the ellipse. If S is the focus, ZZ' is the directrix and P is any point on the ellipse, such that M is the foot of perpendicular from P on ZZ' then

SP = ePM

The equation ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0 represent an ellipse, if

D = abc + 2fgh – af^{2} – bg^{2} – ch^{2} ≠ 0 and h^{2} < ab.

## Standard Form Of Ellipse

The equation of the ellipse whose axes are parallel to the coordinate axes and whose centre is at origin, is

$$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1\space\\\text{with following properties :}$$

$$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, a\gt b$$ | $$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1,a\lt b$$ | |

Coordiantes of the centre | (0, 0) | (0, 0) |

Coordinate of the vertices | (a, 0) and (– a, 0) | (0, – b) and (0, b) |

Coordinate of foci | (ae, 0) and (– ae, 0) | (0, be) and (0, – be) |

Length of major axis | 2a | 2b |

Length of minor axis | 2b | 2a |

Equation of major axis | y = 0 | x = 0 |

Equation of minor axis | x = 0 | y = 0 |

Equation of directrix | $$x =\frac{a}{e}\space\text{and x} =-\frac{a}{e}$$ | $$y =\frac{b}{e}\text{and y}=\frac{-b}{e}$$ |

Eccentricity | $$e =\sqrt{1 -\frac{b^{2}}{a^{2}}}$$ | $$e =\sqrt{1 - \frac{a^{2}}{b^{2}}}$$ |

Length of the latus-rectum | $$\frac{2b^{2}}{a}$$ | $$\frac{2a^{2}}{b}$$ |

Focal distance of point (x, y) | a ± ex | b ± ey |

## Hyperbola

A hyperbola is the set of all points in a plane, the difference of whose distance from two fixed points in a plane is a constant.

The general equation of hyperbola is of the form

ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0

where abc + 2fgh – af^{2} – bg^{2} – ch^{2} ≠ 0 and h^{2 }> ab.

## Standard Form Of Ellipse

The equation of hyperbola having its centre at origin and axis along the coordinate axis

$$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}} = 1\\\text{with following properties :}$$

$$\text{Hyperbola}\space\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$$ | $$\text{Conjugate Hyperbola}\\\frac{-x^{2}}{a^{2}}+\frac{-y^{2}}{b^{2}} = 1$$ | |

Coordiantes of the centre | (0, 0) | (0, 0) |

Coordinate of the vertices | (a, 0) and (– a, 0) | (0, – b) and (0, b) |

Coordinate of foci | (± ae, 0) | (0, ± be) |

Length of transverse axis | 2a | 2b |

Length of conjugate axis | 2b | 2a |

Equation of directriees | $$x =\pm\frac{a}{e}$$ | $$y =\pm\frac{b}{e}$$ |

Eccentricity | $$e =\sqrt{\frac{a^{2} + b^{2}}{a^{2}}}\\\text{or b}^{2}= b^{2}(a^{2}-1)$$ | $$e = \sqrt{\frac{b^{2} + a^{2}}{b^{2}}}\\\text{or}\space a^{2} = b^{2}(e^{2}-1)$$ |

Length of the latus-rectum | $$\frac{2b^{2}}{a}$$ | $$\frac{2a^{2}}{b}$$ |

Equation of the transverse axis | y = 0 | x = 0 |

Equation of the conjugate axis | x = 0 | y = 0 |

## Conjugate Hyperbola

The hyperbola whose transverse and conjugate axis are respectively the conjugate and transverse axes of a given hyperbola is called the conjugate hyperbola of the given hyperbola.

The conjugate hyper of the hyperbola

$$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}} = 1\\\text{is\space}\frac{-x^{2}}{a^{2}} + \frac{-y^{2}}{b^{2}} = 1$$

## General Equation Of Conic Section

^{2}+ 2hxy + by

^{2}+ 2gx + 2fy + c = 0

- A pair of striaght line if ∆ = abc + 2fgh – af
^{2}– bg^{2}– ch^{2}= 0 - Represent a circle if ∆ = abc + 2fgh – af
^{2}– bg^{2}– ch^{2}≠ 0 and h = 0, a = b - Represent a parabola if ∆ = abc + 2fgh – af
^{2}– bg^{2}– ch^{2}≠ 0 and h^{2}= ab - Represent an ellipse if ∆ = abc + 2fgh – af
^{2}– bg^{2}– ch^{2}≠ 0 and h^{2 }< ab - Represent a hyperbola, if ∆ = abc + 2fgh – af
^{2}– bg^{2}– ch^{2}≠ 0 and h^{2}> ab - A rectangular hyperbola if ∆ = abc + 2fgh – af
^{2}– bg^{2}– ch^{2}≠ 0 and a + b = 0

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