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 = 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
- 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
