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Title: Conic Sections
Description: In this part of maths we will learn a lot about Conic sections. This includes: Conic sections, Degenerated conics, Parabolas, Standard equations of parabola, Ellipse, Standard equations of ellipse, Hyperbola and Standard equations of hyperbola. Further details are given in the PDF.
Description: In this part of maths we will learn a lot about Conic sections. This includes: Conic sections, Degenerated conics, Parabolas, Standard equations of parabola, Ellipse, Standard equations of ellipse, Hyperbola and Standard equations of hyperbola. Further details are given in the PDF.
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Conic Sections
Conic sections
Conic sections or conics are the curves that are obtained by intersecting a plane with a
double-napped right circular cone
...
A double-napped cone can be obtained by rotating a line (let us say m) about a fixed vertical
line (let us say l)
...
The intersection (V) of l and m is called the vertex of the cone
...
Condition
θ2 = 90° (Only one nappe of
the cone is entirely cut by the
plane)
Conic
Formed
A circle
Figure
θ1< θ2< 90° (Only one
nappe of the cone is entirely
cut by the plane)
An ellipse
θ1 = θ2 (Only one nappe of
the cone is entirely cut by the
plane)
A parabola
0 < θ2< θ1 (Both the nappes
of the cone are entirely cut by
the plane)
A
hyperbola
Degenerated conics
The conics obtained by cutting a plane with a double-napped cone at its vertex are known as
degenerating conic sections
...
Condition
Conic Formed
Figure
θ2 = 90°
A point
θ1 = θ2
A straight line
0 ≤ θ 2< θ 1
A hyperbola
Parabola
A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed
point (not on the line in the plane)
...
The fixed point F is called the focus
...
The point of intersection of parabola with the axis is called the vertex of the parabola
...
Standard equations of parabola
Open Towards
Right
Standard Equation
y2 = 4ax,
a>0
Coordinates of Focus
(a, 0)
Coordinates of Vertex
(0, 0)
Equation of Directrix
x = –a
Length of Latus Rectum
4a
Axis of Parabola
y=0
Open Towards
Left
Standard Equation
y2 = –
4ax, a >
0
Coordinates of Focus
(–a, 0)
Coordinates of Vertex
(0, 0)
Equation of Directrix
x=a
Length of Latus Rectum 4a
Axis of Parabola
y=0
Open Towards
Upward
Standard Equation
x2 = 4ay, a
>0
Coordinates of Focus
(0, a)
Coordinates of Vertex
(0, 0)
Equation of Directrix
y = –a
Length of Latus
Rectum
4a
Axis of Parabola
x=0
Open Towards
Downward
Standard Equation
x2 = –4ay, a
>0
Coordinates of Focus
(0, –a)
Coordinates of Vertex
(0, 0)
Equation of Directrix
y=a
Length of Latus Rectum
4a
Axis of Parabola
x=0
Example: Consider a parabola x2 = –16y
...
We call this straight line the degenerate case of parabola
...
The two fixed points are called the foci
...
The mid-point of the line segment joining the foci is called the centre of the ellipse
...
The end points of the major axis are called the vertices of the ellipse
...
An ellipse is symmetric with respect to both the coordinate axes
...
Standard equations of ellipse:
Standard Equation
Centre
Vertex
End points of minor
axis
Foci
Length of major axis
Length of minor Axis
Length between foci
Relation between a, b
and c
(0, 0)
(±a, 0)
(0, ±b)
(±c, 0)
2a along x-axis
2b along y-axis
2c along x-axis
Length of latus rectum
Eccentricity (e <1)
Standard Equation
Centre
Vertex
End points of minor
axis
Foci
Length of major axis
Length of minor Axis
Length between foci
Relation between a, b
and c
Length of latus rectum
Eccentricity (e <1)
Example: Consider the ellipse 9x2 + 4y2 = 36
...
(0, 0)
(0, ±a)
(±b, 0)
(0, ±c)
2a along y-axis
2b along x-axis
2c along y-axis
a 2 = b 2 + c2
⇒ c2 = (3)2 – (2)2 = 9 – 4 = 5
Eccentricity,
Coordinates of the centre = (0, 0)
Coordinates of the vertices = (0, ±a) = (0, ±3)
Coordinates of the foci = (0, ± c) =
Length of the major axis = 2a = 2 × 3 = 6
Length of the minor axis = 2b = 2 × 2 = 4
Hyperbola
A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed
points in the plane is a constant
...
The constant, which is the difference of the distances of a point on the hyperbola from
the two fixed points, is always less than the distance between the two fixed points
...
The line through the foci is called the transverse axis and the line through the centre
and perpendicular to the transverse axis is called the conjugate axis
...
The eccentricity of the hyperbola is the ratio of the distances from the centre of the
hyperbola to one of the foci and to one of the vertices of the hyperbola
...
The line segment that is perpendicular to the transverse axis through the focus and
whose end points lie on the hyperbola is called the latus rectum of the hyperbola
...
16x2 – 9y2 = 144 can be rewritten as
Comparing with standard form of hyperbola
Here, a = 3 and b = 4
...
(0, 0)
(0, ±a)
(0, ±c)
x-axis
y-axis
2b
2a
2c
Coordinates of foci = (±c, 0) = (±5, 0)
Length of transverse axis = 2a = 2 × 3 = 6
Length of conjugate axis = 2b = 2 × 4 = 8
Distance between foci = 2c = 2 × 5 = 10
Length of latus rectum
A hyperbola having equal lengths of both the axes i
...
, transverse and conjugate (a = b) is
called an equilateral hyperbola
Title: Conic Sections
Description: In this part of maths we will learn a lot about Conic sections. This includes: Conic sections, Degenerated conics, Parabolas, Standard equations of parabola, Ellipse, Standard equations of ellipse, Hyperbola and Standard equations of hyperbola. Further details are given in the PDF.
Description: In this part of maths we will learn a lot about Conic sections. This includes: Conic sections, Degenerated conics, Parabolas, Standard equations of parabola, Ellipse, Standard equations of ellipse, Hyperbola and Standard equations of hyperbola. Further details are given in the PDF.