NCERT Textbook: Conic Sections

# NCERT Textbook: Conic Sections | NCERT Textbooks (Class 6 to Class 12) - CTET & State TET PDF Download

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176    MATHEMATICS
vLet the relation of knowledge to real life be very visible to your pupils
and let them understand how by knowledge the world could be
transformed. – BERTRAND RUSSELL v
10.1 Introduction
In the preceding Chapter 10, we have studied various forms
of the equations of a line. In this Chapter, we shall study
about some other curves, viz., circles, ellipses, parabolas
and hyperbolas. The names parabola and hyperbola are
given by Apollonius. These curves are in fact, known as
conic sections or more commonly conics because they
can be obtained as intersections of a plane with a double
napped right circular cone. These curves have a very wide
range of applications in fields such as planetary motion,
design of telescopes and antennas, reflectors in flashlights
and automobile headlights, etc.  Now, in the subsequent sections we will see how the
intersection of a plane with a double napped right circular cone
results in different types of curves.
10.2 Sections of a Cone
Let l be a fixed vertical line and m be another line intersecting it at
a fixed point V and inclined to it at an angle a (Fig10.1).
Suppose we rotate the line m around the line l in such a way
that the angle a remains constant.  Then the surface generated is
a double-napped right circular hollow cone herein after referred as
Apollonius
(262 B.C. -190 B.C.)
10 Chapter
Fig 10. 1
CONIC SECTIONS
2024-25
Page 2

176    MATHEMATICS
vLet the relation of knowledge to real life be very visible to your pupils
and let them understand how by knowledge the world could be
transformed. – BERTRAND RUSSELL v
10.1 Introduction
In the preceding Chapter 10, we have studied various forms
of the equations of a line. In this Chapter, we shall study
about some other curves, viz., circles, ellipses, parabolas
and hyperbolas. The names parabola and hyperbola are
given by Apollonius. These curves are in fact, known as
conic sections or more commonly conics because they
can be obtained as intersections of a plane with a double
napped right circular cone. These curves have a very wide
range of applications in fields such as planetary motion,
design of telescopes and antennas, reflectors in flashlights
and automobile headlights, etc.  Now, in the subsequent sections we will see how the
intersection of a plane with a double napped right circular cone
results in different types of curves.
10.2 Sections of a Cone
Let l be a fixed vertical line and m be another line intersecting it at
a fixed point V and inclined to it at an angle a (Fig10.1).
Suppose we rotate the line m around the line l in such a way
that the angle a remains constant.  Then the surface generated is
a double-napped right circular hollow cone herein after referred as
Apollonius
(262 B.C. -190 B.C.)
10 Chapter
Fig 10. 1
CONIC SECTIONS
2024-25
CONIC SECTIONS       177
Fig 10. 2 Fig 10. 3
cone and extending indefinitely far in both directions (Fig10.2).
The point V is called the vertex; the line l is the axis of the cone. The rotating line
m is called a generator of the cone.  The vertex separates the cone into two parts
called nappes.
If we take the intersection of a plane with a cone, the section so obtained is called
a conic section. Thus, conic sections are the curves obtained by intersecting a right
circular cone by a plane.
We obtain different kinds of conic sections depending on the position of the
intersecting plane with respect to the cone and by the angle made by it with the vertical
axis of the cone. Let ß be the angle made by the intersecting plane with the vertical
axis of the cone (Fig10.3).
The intersection of the plane with the cone can take place either at the vertex of
the cone or at any other part of the nappe either below or above the vertex.
10.2.1 Circle, ellipse, parabola and hyperbola When the plane cuts the nappe (other
than the vertex) of the cone, we have the following situations:
(a) When ß = 90
o
,  the section is a circle (Fig10.4).
(b) When a < ß <  90
o
, the section is an ellipse (Fig10.5).
(c) When ß = a; the section is a parabola (Fig10.6).
(In each of the above three situations, the plane cuts entirely across one nappe of
the cone).
(d) When 0 = ß < a; the plane cuts through both the nappes and the curves of
intersection is a hyperbola (Fig10.7).
2024-25
Page 3

176    MATHEMATICS
vLet the relation of knowledge to real life be very visible to your pupils
and let them understand how by knowledge the world could be
transformed. – BERTRAND RUSSELL v
10.1 Introduction
In the preceding Chapter 10, we have studied various forms
of the equations of a line. In this Chapter, we shall study
about some other curves, viz., circles, ellipses, parabolas
and hyperbolas. The names parabola and hyperbola are
given by Apollonius. These curves are in fact, known as
conic sections or more commonly conics because they
can be obtained as intersections of a plane with a double
napped right circular cone. These curves have a very wide
range of applications in fields such as planetary motion,
design of telescopes and antennas, reflectors in flashlights
and automobile headlights, etc.  Now, in the subsequent sections we will see how the
intersection of a plane with a double napped right circular cone
results in different types of curves.
10.2 Sections of a Cone
Let l be a fixed vertical line and m be another line intersecting it at
a fixed point V and inclined to it at an angle a (Fig10.1).
Suppose we rotate the line m around the line l in such a way
that the angle a remains constant.  Then the surface generated is
a double-napped right circular hollow cone herein after referred as
Apollonius
(262 B.C. -190 B.C.)
10 Chapter
Fig 10. 1
CONIC SECTIONS
2024-25
CONIC SECTIONS       177
Fig 10. 2 Fig 10. 3
cone and extending indefinitely far in both directions (Fig10.2).
The point V is called the vertex; the line l is the axis of the cone. The rotating line
m is called a generator of the cone.  The vertex separates the cone into two parts
called nappes.
If we take the intersection of a plane with a cone, the section so obtained is called
a conic section. Thus, conic sections are the curves obtained by intersecting a right
circular cone by a plane.
We obtain different kinds of conic sections depending on the position of the
intersecting plane with respect to the cone and by the angle made by it with the vertical
axis of the cone. Let ß be the angle made by the intersecting plane with the vertical
axis of the cone (Fig10.3).
The intersection of the plane with the cone can take place either at the vertex of
the cone or at any other part of the nappe either below or above the vertex.
10.2.1 Circle, ellipse, parabola and hyperbola When the plane cuts the nappe (other
than the vertex) of the cone, we have the following situations:
(a) When ß = 90
o
,  the section is a circle (Fig10.4).
(b) When a < ß <  90
o
, the section is an ellipse (Fig10.5).
(c) When ß = a; the section is a parabola (Fig10.6).
(In each of the above three situations, the plane cuts entirely across one nappe of
the cone).
(d) When 0 = ß < a; the plane cuts through both the nappes and the curves of
intersection is a hyperbola (Fig10.7).
2024-25
178    MATHEMATICS
Fig 10. 4
10.2.2  Degenerated conic sections
When the plane cuts at the vertex of the cone, we have  the following  different cases:
(a) When a < ß = 90
o
, then the section is a point (Fig10.8).
(b) When ß = a, the plane contains a generator of the cone and the section is a
straight line  (Fig10.9).
It is the degenerated case of a parabola.
(c) When 0 = ß < a, the section is a pair of intersecting straight lines (Fig10.10).
It is the degenerated case of a hyperbola.
Fig 10. 6
Fig 10. 7
Fig 10. 5
2024-25
Page 4

176    MATHEMATICS
vLet the relation of knowledge to real life be very visible to your pupils
and let them understand how by knowledge the world could be
transformed. – BERTRAND RUSSELL v
10.1 Introduction
In the preceding Chapter 10, we have studied various forms
of the equations of a line. In this Chapter, we shall study
about some other curves, viz., circles, ellipses, parabolas
and hyperbolas. The names parabola and hyperbola are
given by Apollonius. These curves are in fact, known as
conic sections or more commonly conics because they
can be obtained as intersections of a plane with a double
napped right circular cone. These curves have a very wide
range of applications in fields such as planetary motion,
design of telescopes and antennas, reflectors in flashlights
and automobile headlights, etc.  Now, in the subsequent sections we will see how the
intersection of a plane with a double napped right circular cone
results in different types of curves.
10.2 Sections of a Cone
Let l be a fixed vertical line and m be another line intersecting it at
a fixed point V and inclined to it at an angle a (Fig10.1).
Suppose we rotate the line m around the line l in such a way
that the angle a remains constant.  Then the surface generated is
a double-napped right circular hollow cone herein after referred as
Apollonius
(262 B.C. -190 B.C.)
10 Chapter
Fig 10. 1
CONIC SECTIONS
2024-25
CONIC SECTIONS       177
Fig 10. 2 Fig 10. 3
cone and extending indefinitely far in both directions (Fig10.2).
The point V is called the vertex; the line l is the axis of the cone. The rotating line
m is called a generator of the cone.  The vertex separates the cone into two parts
called nappes.
If we take the intersection of a plane with a cone, the section so obtained is called
a conic section. Thus, conic sections are the curves obtained by intersecting a right
circular cone by a plane.
We obtain different kinds of conic sections depending on the position of the
intersecting plane with respect to the cone and by the angle made by it with the vertical
axis of the cone. Let ß be the angle made by the intersecting plane with the vertical
axis of the cone (Fig10.3).
The intersection of the plane with the cone can take place either at the vertex of
the cone or at any other part of the nappe either below or above the vertex.
10.2.1 Circle, ellipse, parabola and hyperbola When the plane cuts the nappe (other
than the vertex) of the cone, we have the following situations:
(a) When ß = 90
o
,  the section is a circle (Fig10.4).
(b) When a < ß <  90
o
, the section is an ellipse (Fig10.5).
(c) When ß = a; the section is a parabola (Fig10.6).
(In each of the above three situations, the plane cuts entirely across one nappe of
the cone).
(d) When 0 = ß < a; the plane cuts through both the nappes and the curves of
intersection is a hyperbola (Fig10.7).
2024-25
178    MATHEMATICS
Fig 10. 4
10.2.2  Degenerated conic sections
When the plane cuts at the vertex of the cone, we have  the following  different cases:
(a) When a < ß = 90
o
, then the section is a point (Fig10.8).
(b) When ß = a, the plane contains a generator of the cone and the section is a
straight line  (Fig10.9).
It is the degenerated case of a parabola.
(c) When 0 = ß < a, the section is a pair of intersecting straight lines (Fig10.10).
It is the degenerated case of a hyperbola.
Fig 10. 6
Fig 10. 7
Fig 10. 5
2024-25
CONIC SECTIONS       179
In the following sections, we shall obtain the equations of each of  these conic
sections in standard form  by defining them based on geometric properties.
Fig 10. 8
Fig 10. 9
Fig 10. 10
10.3 Circle
Definition 1 A circle is the set of all points in a plane that are equidistant from a fixed
point in the plane.
The fixed point is called the centre of the circle and the distance from the centre
to a point on the circle is called the radius of the circle (Fig 10.11).
2024-25
Page 5

176    MATHEMATICS
vLet the relation of knowledge to real life be very visible to your pupils
and let them understand how by knowledge the world could be
transformed. – BERTRAND RUSSELL v
10.1 Introduction
In the preceding Chapter 10, we have studied various forms
of the equations of a line. In this Chapter, we shall study
about some other curves, viz., circles, ellipses, parabolas
and hyperbolas. The names parabola and hyperbola are
given by Apollonius. These curves are in fact, known as
conic sections or more commonly conics because they
can be obtained as intersections of a plane with a double
napped right circular cone. These curves have a very wide
range of applications in fields such as planetary motion,
design of telescopes and antennas, reflectors in flashlights
and automobile headlights, etc.  Now, in the subsequent sections we will see how the
intersection of a plane with a double napped right circular cone
results in different types of curves.
10.2 Sections of a Cone
Let l be a fixed vertical line and m be another line intersecting it at
a fixed point V and inclined to it at an angle a (Fig10.1).
Suppose we rotate the line m around the line l in such a way
that the angle a remains constant.  Then the surface generated is
a double-napped right circular hollow cone herein after referred as
Apollonius
(262 B.C. -190 B.C.)
10 Chapter
Fig 10. 1
CONIC SECTIONS
2024-25
CONIC SECTIONS       177
Fig 10. 2 Fig 10. 3
cone and extending indefinitely far in both directions (Fig10.2).
The point V is called the vertex; the line l is the axis of the cone. The rotating line
m is called a generator of the cone.  The vertex separates the cone into two parts
called nappes.
If we take the intersection of a plane with a cone, the section so obtained is called
a conic section. Thus, conic sections are the curves obtained by intersecting a right
circular cone by a plane.
We obtain different kinds of conic sections depending on the position of the
intersecting plane with respect to the cone and by the angle made by it with the vertical
axis of the cone. Let ß be the angle made by the intersecting plane with the vertical
axis of the cone (Fig10.3).
The intersection of the plane with the cone can take place either at the vertex of
the cone or at any other part of the nappe either below or above the vertex.
10.2.1 Circle, ellipse, parabola and hyperbola When the plane cuts the nappe (other
than the vertex) of the cone, we have the following situations:
(a) When ß = 90
o
,  the section is a circle (Fig10.4).
(b) When a < ß <  90
o
, the section is an ellipse (Fig10.5).
(c) When ß = a; the section is a parabola (Fig10.6).
(In each of the above three situations, the plane cuts entirely across one nappe of
the cone).
(d) When 0 = ß < a; the plane cuts through both the nappes and the curves of
intersection is a hyperbola (Fig10.7).
2024-25
178    MATHEMATICS
Fig 10. 4
10.2.2  Degenerated conic sections
When the plane cuts at the vertex of the cone, we have  the following  different cases:
(a) When a < ß = 90
o
, then the section is a point (Fig10.8).
(b) When ß = a, the plane contains a generator of the cone and the section is a
straight line  (Fig10.9).
It is the degenerated case of a parabola.
(c) When 0 = ß < a, the section is a pair of intersecting straight lines (Fig10.10).
It is the degenerated case of a hyperbola.
Fig 10. 6
Fig 10. 7
Fig 10. 5
2024-25
CONIC SECTIONS       179
In the following sections, we shall obtain the equations of each of  these conic
sections in standard form  by defining them based on geometric properties.
Fig 10. 8
Fig 10. 9
Fig 10. 10
10.3 Circle
Definition 1 A circle is the set of all points in a plane that are equidistant from a fixed
point in the plane.
The fixed point is called the centre of the circle and the distance from the centre
to a point on the circle is called the radius of the circle (Fig 10.11).
2024-25
180    MATHEMATICS
The equation of the circle is simplest if the centre of the circle is at the origin.
However, we derive below the equation of the circle with a given centre and radius
(Fig 10.12).
Given C (h, k) be the centre and r the radius of circle. Let P(x, y) be any point on
the circle (Fig10.12).  Then,  by the definition, | CP | = r . By the distance formula,
we have
2 2
( ) ( ) x – h y – k r + =
i.e. (x – h)
2
+ (y – k)
2
= r
2
This is the required equation of the circle with centre at (h,k) and radius r .
Example 1  Find an equation of the circle with centre at (0,0) and radius r.
Solution Here  h = k = 0. Therefore, the equation of the circle is x
2
+ y
2
= r
2
.
Example 2 Find the equation of the circle with centre (–3, 2) and radius 4.
Solution Here h = –3, k = 2 and r = 4. Therefore, the equation of the required circle is
(x + 3)
2
+ (y –2)
2
= 16
Example 3 Find the centre and the radius of the circle x
2
+ y
2
+ 8x + 10y – 8 = 0
Solution  The given equation is
(x
2
+ 8x)  + (y
2
+ 10y) = 8
Now, completing the squares within the parenthesis, we get
(x
2
+ 8x + 16)  +  (y
2
+ 10y + 25)  = 8 + 16 + 25
i.e. (x + 4)
2
+ (y + 5)
2
= 49
i.e. {x – (– 4)}
2
+ {y – (–5)}
2
= 7
2
Therefore, the given circle has centre at (– 4, –5) and radius 7.
Fig 10. 11
Fig 10. 12
2024-25
```

## NCERT Textbooks (Class 6 to Class 12)

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## FAQs on NCERT Textbook: Conic Sections - NCERT Textbooks (Class 6 to Class 12) - CTET & State TET

 1. What are conic sections in mathematics?
Ans. Conic sections are the curves obtained by intersecting a cone with a plane. They include the shapes of a circle, ellipse, parabola, and hyperbola.
 2. How are conic sections useful in real life?
Ans. Conic sections have various applications in real life. For example, they are used in designing satellite dish antennas (parabolas), modeling the orbits of planets (ellipses), analyzing the trajectory of projectiles (parabolas), and understanding the shapes of natural phenomena like rainbows and eclipses.
 3. What is the equation of a circle in conic sections?
Ans. The equation of a circle in conic sections is given by (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle and r represents the radius.
 4. How can conic sections be classified based on their eccentricity?
Ans. Conic sections can be classified into three types based on their eccentricity: circles (eccentricity = 0), ellipses (0 < eccentricity < 1), and hyperbolas (eccentricity > 1). Parabolas are a special case with eccentricity equal to 1.
 5. What is the focus-directrix property of conic sections?
Ans. The focus-directrix property is a fundamental property of conic sections. It states that for any point on a conic section, the distance to the focus is always equal to the perpendicular distance to the directrix. This property helps in defining and understanding the geometric properties of conic sections.

## NCERT Textbooks (Class 6 to Class 12)

678 docs|672 tests

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