# Conic Sections Class 11 Notes Mathematics Chapter 11 - CBSE

## 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 = a2

## General Equation Of A Circle

The general equation of a circle is

x2 + y2 + 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 (x1 , y1) and (x2 , y2) as diameter is (x – x1) (x – x2) + (y – y1) (y – y2) = 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

ax2 + 2hxy + by2+ 2gx + 2fy + c = 0,

satisfying the condition

abc + 2fgh – af2 – bg2 – ch2 ≠ 0 and h2 = ab. Standard form of a parabola is

y2 = 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

 y2 = 4ax y2 = -4ax x2 = 4ay x2 = 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 ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represent an ellipse, if

D = abc + 2fgh – af2 – bg2 – ch2 ≠ 0 and h2 < 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

ax2 + 2hxy + by2 + 2gx + 2fy + c = 0

where abc + 2fgh – af2 – bg2 – ch2 ≠ 0 and h2 > 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

The general equation of second degree of the conic section is ax2 + 2hxy + by2 + 2gx + 2fy + c = 0
• A pair of striaght line if ∆ = abc + 2fgh – af2 – bg2 – ch2 = 0
• Represent a circle if ∆ = abc + 2fgh – af2 – bg2 – ch2 ≠ 0 and h = 0, a = b
• Represent a parabola if ∆ = abc + 2fgh – af2 – bg2 – ch2 ≠ 0 and h2 = ab
• Represent an ellipse if ∆ = abc + 2fgh – af2 – bg2 – ch2 ≠ 0 and h2 < ab
• Represent a hyperbola, if ∆ = abc + 2fgh – af2 – bg2 – ch2 ≠ 0 and h2 > ab
• A rectangular hyperbola if ∆ = abc + 2fgh – af2 – bg2 – ch2 ≠ 0 and a + b = 0