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NCERT Class 11 Maths Textbook Chapter 11 With Answer Book PDF Free Download
Chapter 11: Conic Sections
11.1 Introduction
In the preceding Chapter 10, we have studied various forms of the equations of a line. In this chapter, we shall study 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.
11.3 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 center of the circle and the distance from the center to a point on the circle is called the radius of the circle.
11.4 Parabola
Definition 2 A parabola is the set of all points in a plane that is equidistant from a fixed line and a fixed point (not on the line) in the plane.
| Author | NCERT |
| Language | English |
| No. of Pages | 32 |
| PDF Size | 2.4 MB |
| Category | Mathematics |
| Source/Credits | ncert.nic.in |
NCERT Solutions Class 11 Maths Chapter 11 Conic Sections
In each of the following Exercises 1 to 5, find the equation of the circle with
1. Centre (0, 2) and radius 2
Solution:
Given:
Centre (0, 2) and radius 2
Let us consider the equation of a circle with centre (h, k) and
Radius r is given as (x – h)2 + (y – k)2 = r2
So, centre (h, k) = (0, 2) and radius (r) = 2
The equation of the circle is
(x – 0)2 + (y – 2)2 = 22
x2 + y2 + 4 – 4y = 4
x2 + y2 – 4y = 0
∴ The equation of the circle is x2 + y2 – 4y = 0
2. Centre (–2, 3) and radius 4
Solution:
Given:
Centre (-2, 3) and radius 4
Let us consider the equation of a circle with centre (h, k) and
Radius r is given as (x – h)2 + (y – k)2 = r2
So, centre (h, k) = (-2, 3) and radius (r) = 4
The equation of the circle is
(x + 2)2 + (y – 3)2 = (4)2
x2 + 4x + 4 + y2 – 6y + 9 = 16
x2 + y2 + 4x – 6y – 3 = 0
∴ The equation of the circle is x2 + y2 + 4x – 6y – 3 = 0
3. Centre (1/2, 1/4) and radius (1/12)
Solution:
Given:
Centre (1/2, 1/4) and radius 1/12
Let us consider the equation of a circle with center (h, k) and
Radius r is given as (x – h)2 + (y – k)2 = r2
So, centre (h, k) = (1/2, 1/4) and radius (r) = 1/12
The equation of the circle is
(x – 1/2)2 + (y – 1/4)2 = (1/12)2
x2 – x + ¼ + y2 – y/2 + 1/16 = 1/144
x2 – x + ¼ + y2 – y/2 + 1/16 = 1/144
144x2 – 144x + 36 + 144y2 – 72y + 9 – 1 = 0
144x2 – 144x + 144y2 – 72y + 44 = 0
36x2 + 36x + 36y2 – 18y + 11 = 0
36x2 + 36y2 – 36x – 18y + 11= 0
∴ The equation of the circle is 36x2 + 36y2 – 36x – 18y + 11= 0
4. Centre (1, 1) and radius √2
Solution:
Given:
Centre (1, 1) and radius √2
Let us consider the equation of a circle with centre (h, k) and
Radius r is given as (x – h)2 + (y – k)2 = r2
So, centre (h, k) = (1, 1) and radius (r) = √2
The equation of the circle is
(x-1)2 + (y-1)2 = (√2)2
x2 – 2x + 1 + y2 -2y + 1 = 2
x2 + y2 – 2x -2y = 0
∴ The equation of the circle is x2 + y2 – 2x -2y = 0
5. Centre (–a, –b) and radius √(a2 – b2)
Solution:
Given:
Centre (-a, -b) and radius √(a2 – b2)
Let us consider the equation of a circle with centre (h, k) and
Radius r is given as (x – h)2 + (y – k)2 = r2
So, centre (h, k) = (-a, -b) and radius (r) = √(a2 – b2)
The equation of the circle is
(x + a)2 + (y + b)2 = (√(a2 – b2)2)
x2 + 2ax + a2 + y2 + 2by + b2 = a2 – b2
x2 + y2 +2ax + 2by + 2b2 = 0
∴ The equation of the circle is x2 + y2 +2ax + 2by + 2b2 = 0
In each of the following Exercise 6 to 9, find the centre and radius of the circles.
6. (x + 5)2 + (y – 3)2 = 36
Solution:
Given:
The equation of the given circle is (x + 5)2 + (y – 3)2 = 36
(x – (-5))2 + (y – 3)2 = 62 [which is of the form (x – h)2 + (y – k )2 = r2]
Where, h = -5, k = 3 and r = 6
∴ The centre of the given circle is (-5, 3) and its radius is 6.
7. x2 + y2 – 4x – 8y – 45 = 0
Solution:
Given:
The equation of the given circle is x2 + y2 – 4x – 8y – 45 = 0.
x2 + y2 – 4x – 8y – 45 = 0
(x2 – 4x) + (y2 -8y) = 45
(x2 – 2(x) (2) + 22) + (y2 – 2(y) (4) + 42) – 4 – 16 = 45
(x – 2)2 + (y – 4)2 = 65
(x – 2)2 + (y – 4)2 = (√65)2 [which is form (x-h)2 +(y-k)2 = r2]
Where h = 2, K = 4 and r = √65
∴ The centre of the given circle is (2, 4) and its radius is √65.
8. x2 + y2 – 8x + 10y – 12 = 0
Solution:
Given:
The equation of the given circle is x2 + y2 -8x + 10y -12 = 0.
x2 + y2 – 8x + 10y – 12 = 0
(x2 – 8x) + (y2 + 10y) = 12
(x2 – 2(x) (4) + 42) + (y2 – 2(y) (5) + 52) – 16 – 25 = 12
(x – 4)2 + (y + 5)2 = 53
(x – 4)2 + (y – (-5))2 = (√53)2 [which is form (x-h)2 +(y-k)2 = r2]
Where h = 4, K= -5 and r = √53
∴ The centre of the given circle is (4, -5) and its radius is √53.
9. 2x2 + 2y2 – x = 0
Solution:
The equation of the given of the circle is 2x2 + 2y2 –x = 0.
2x2 + 2y2 –x = 0
(2x2 + x) + 2y2 = 0
(x2 – 2 (x) (1/4) + (1/4)2) + y2 – (1/4)2 = 0
(x – 1/4)2 + (y – 0)2 = (1/4)2 [which is form (x-h)2 +(y-k)2 = r2]
Where, h = ¼, K = 0, and r = ¼
∴ The center of the given circle is (1/4, 0) and its radius is 1/4.
10. Find the equation of the circle passing through the points (4,1) and (6,5) and whose center is on the line 4x + y = 16.
Solution:
Let us consider the equation of the required circle be (x – h)2+ (y – k)2 = r2
We know that the circle passes through points (4,1) and (6,5)
So,
(4 – h)2 + (1 – k)2 = r2 ……………..(1)
(6– h)2+ (5 – k)2 = r2 ………………(2)
Since, the centre (h, k) of the circle lies on line 4x + y = 16,
4h + k =16………………… (3)
From the equation (1) and (2), we obtain
(4 – h)2+ (1 – k)2 =(6 – h)2 + (5 – k)2
16 – 8h + h2 +1 -2k +k2 = 36 -12h +h2+15 – 10k + k2
16 – 8h +1 -2k + 12h -25 -10k
4h +8k = 44
h + 2k =11……………. (4)
On solving equations (3) and (4), we obtain h=3 and k= 4.
On substituting the values of h and k in equation (1), we obtain
(4 – 3)2+ (1 – 4)2 = r2
(1)2 + (-3)2 = r2
1+9 = r2
r = √10
so now, (x – 3)2 + (y – 4)2 = (√10)2
x2 – 6x + 9 + y2 – 8y + 16 =10
x2 + y2 – 6x – 8y + 15 = 0
∴ The equation of the required circle is x2 + y2 – 6x – 8y + 15 = 0
NCERT Class 11 Maths Textbook Chapter 11 With Answer Book PDF Free Download