Chapter 23 : The Straight Lines - Rd Sharma Solutions for Class 11-science Maths CBSE

Chapter 23 - The Straight Lines Exercise Ex. 23.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Chapter 23 - The Straight Lines Exercise Ex. 23.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Chapter 23 - The Straight Lines Exercise Ex. 23.3

Question 1

Solution 1

Question 2

Solution 2

y minus y subscript 1 equals m open parentheses x minus x subscript 1 close parentheses
S i n c e space t h e space l i n e space c u t s space t h e space x minus a x i s space a t space open parentheses minus 3 comma 0 close parentheses space w i t h space s l o p e space minus 2 comma space w e space h a v e comma
y minus 0 equals minus 2 open parentheses x plus 3 close parentheses
rightwards double arrow y equals minus 2 x minus 6
rightwards double arrow 2 x plus y plus 6 equals 0

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Chapter 23 - The Straight Lines Exercise Ex. 23.4

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

T h e space l i n e space p a s s e s space t h r o u g h space t h e space p o i n t space open parentheses 2 comma 0 close parentheses.
A l s o space i t s space i n c l i n a t i o n space t o space y minus a x i s space i s space 135 degree.
T h a t space i s comma space t h e space i n c l i n a t i o n space o f space t h e space g i v e n space l i n e space w i t h space t h e space x minus a x i s space i s space 180 degree minus 135 degree.
T h a t space i s comma space t h e space s l o p e space o f space t h e space g i v e n space l i n e space i s space 45 degree
T h e space e q u a t i o n space o f space t h e space l i n e space h a v i n g space s l o p e space apostrophe m apostrophe space a n d space p a s sin g space t h r o u g h space t h e
p o i n t space open parentheses x subscript 1 comma y subscript 1 close parentheses space i s space y minus y subscript 1 equals m open parentheses x minus x subscript 1 close parentheses
T h e r e f o r e comma space t h e space r e q u i r e d space e q u a t i o n space i s
y minus 0 equals tan 45 degree open parentheses x minus 2 close parentheses
rightwards double arrow y equals 1 cross times open parentheses x minus 2 close parentheses
rightwards double arrow y equals x minus 2
rightwards double arrow x minus y minus 2 equals 0

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Chapter 23 - The Straight Lines Exercise Ex. 23.5

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

T h e space r e c tan g l e space A B C D space w i l l space h a v e space d i a g o n a l s space A C space a n d space B D
A C space p a s s e s space t h r o u g h space A open parentheses a comma b close parentheses space a n d space C open parentheses a apostrophe comma b apostrophe close parentheses. space
T h u s space e q u a t i o n space o f space A C space i s :
fraction numerator y minus y subscript 1 over denominator y subscript 2 minus y subscript 1 end fraction equals fraction numerator x minus x subscript 1 over denominator x subscript 2 minus x subscript 1 end fraction
rightwards double arrow fraction numerator y minus b over denominator b apostrophe minus b end fraction equals fraction numerator x minus a over denominator a apostrophe minus a end fraction
rightwards double arrow open parentheses y minus b close parentheses open parentheses a apostrophe minus a close parentheses equals open parentheses x minus a close parentheses open parentheses b apostrophe minus b close parentheses
rightwards double arrow y open parentheses a apostrophe minus a close parentheses minus a apostrophe b plus a b equals x open parentheses b apostrophe minus b close parentheses minus a b apostrophe plus a b
rightwards double arrow y open parentheses a apostrophe minus a close parentheses equals x open parentheses b apostrophe minus b close parentheses minus a b apostrophe plus a apostrophe b
rightwards double arrow y open parentheses a apostrophe minus a close parentheses minus x open parentheses b apostrophe minus b close parentheses equals a apostrophe b minus a b apostrophe

B D space p a s s e s space t h r o u g h space B open parentheses a apostrophe comma b close parentheses space a n d space D open parentheses a comma b apostrophe close parentheses. space
T h u s space e q u a t i o n space o f space B D space i s :
fraction numerator y minus y subscript 1 over denominator y subscript 2 minus y subscript 1 end fraction equals fraction numerator x minus x subscript 1 over denominator x subscript 2 minus x subscript 1 end fraction
rightwards double arrow fraction numerator y minus b over denominator b apostrophe minus b end fraction equals fraction numerator x minus a apostrophe over denominator a minus a apostrophe end fraction
rightwards double arrow open parentheses y minus b close parentheses open parentheses a minus a apostrophe close parentheses equals open parentheses x minus a apostrophe close parentheses open parentheses b apostrophe minus b close parentheses
rightwards double arrow minus y open parentheses a apostrophe minus a close parentheses minus a b plus a apostrophe b equals x open parentheses b apostrophe minus b close parentheses minus a apostrophe b apostrophe plus a apostrophe b
rightwards double arrow a apostrophe b apostrophe minus a b equals x open parentheses b apostrophe minus b close parentheses plus y open parentheses a apostrophe minus a close parentheses
rightwards double arrow x open parentheses b apostrophe minus b close parentheses plus y open parentheses a apostrophe minus a close parentheses equals a apostrophe b apostrophe minus a b

Question 11

Find the equation of the side BC of the triangle ABC whose vertices are A (-1, -2), B (0, 1) and (2, 0) respectively. Also, find the equation of the median through (-1, -2).

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

L equals 4 over 1875 C plus 124.942 minus 4 cross times 20 over 1875
rightwards double arrow L equals 4 over 1875 C plus 124.899

Question 18
Solution 18
Question 19

Solution 19

rightwards double arrow y minus 3 equals 1 third open parentheses x minus 4 close parentheses
rightwards double arrow 3 open parentheses y minus 3 close parentheses equals x minus 4
rightwards double arrow x minus 3 y plus 9 minus 4 equals 0
rightwards double arrow x minus 3 y plus 5 equals 0

Question 20

Solution 20

Question 21

Find the equations of the diagonals of the square formed by the lines x = 0, y = 0, x = 1 and y = 1.

Solution 21

  

 

 

  

Chapter 23 - The Straight Lines Exercise Ex. 23.6

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

3

Question 4

For what values of a and b the intercepts cut off on the coordinate axes by the line ax + by +8 = 0 are equal in length but opposite in signs to those cut off by the line 2x – 3y + 6 = 0 on the axes.

Solution 4


 

 

 

Question 5

Solution 5

Question 6

Find the equation of the line which passing through the point (-4, 3) and the portion of the line intercepted  between the axes is divided internally in the ratio 5:3 by this point

Solution 6

 

 

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

P o i n t space left parenthesis h comma k right parenthesis space d i v i d e s space t h e space l i n e space s e g m e n t space i n space t h e space r a t i o space 1 : 2
T h u s comma space u sin g space s e c t i o n space p o i n t space f o r m u l a comma space w e space h a v e
h equals fraction numerator 2 cross times a plus 1 cross times 0 over denominator 1 plus 2 end fraction
a n d
k equals fraction numerator 2 cross times 0 plus 1 cross times b over denominator 1 plus 2 end fraction
T h e r e f o r e comma space w e space h a v e comma
h equals fraction numerator 2 a over denominator 3 end fraction space a n d space k equals b over 3
rightwards double arrow a equals fraction numerator 3 h over denominator 2 end fraction a n d space b equals 3 k
T h u s comma space t h e space c o r r e s p o n d i n g space p o i n t s space o f space A space a n d space B space a r e space open parentheses fraction numerator 3 h over denominator 2 end fraction comma 0 close parentheses space a n d space open parentheses 0 comma 3 k close parentheses
T h u s comma space t h e space e q u a t i o n space o f space t h e space l i n e space j o i n i n g space t h e space p o i n t s space A space a n d space B space i s
fraction numerator y minus 3 k over denominator 3 k minus 0 end fraction equals fraction numerator x minus 0 over denominator 0 minus fraction numerator 3 h over denominator 2 end fraction end fraction
rightwards double arrow minus fraction numerator 3 h over denominator 2 end fraction open parentheses y minus 3 k close parentheses equals x cross times 3 k
rightwards double arrow minus 3 h y plus 9 h k equals 6 k x
rightwards double arrow 2 k x plus h y equals 3 k h

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Chapter 23 - The Straight Lines Exercise Ex. 23.7

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Find the equation of the line on which the length of the perpendicular segment from the origin to the line is 4 and the inclination of the perpendicular segment with the positive direction of x-axis is 300.

Solution 5


 

 

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Chapter 23 - The Straight Lines Exercise Ex. 23.8

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Chapter 23 - The Straight Lines Exercise Ex. 23.9

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Chapter 23 - The Straight Lines Exercise Ex. 23.10

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

 
Question 17

Solution 17

Question 18

Solution 18

 
Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

 
Question 22

Solution 22

Question 23

Find the equations of the line passing through the intersection of the lines 2x + y = 5 and x + 3y + 8 = 0 and parallel to the line 3x + 4y = 7.

Solution 23


 

 

Question 24

Find the equation of the line passing through the point of intersection of the lines 5x - 6y - 1 = 0 and 3x + 2y + 5 = 0 and perpendicular to the line 3x - 5y + 11 = 0.

Solution 24

  

Chapter 23 - The Straight Lines Exercise Ex. 23.11

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Chapter 23 - The Straight Lines Exercise Ex. 23.12

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

 

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26


Question 27

Find the equation of the straight line passing through the point of intersection of the lines 5x – 6y – 1 = 0 and 3x + 2y + 5 = 0 and perpendicular to the line 3x – 5y + 11 = 0.

Solution 27


 

 

Chapter 23 - The Straight Lines Exercise Ex. 23.13

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14


 

 

Chapter 23 - The Straight Lines Exercise Ex. 23.14

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Chapter 23 - The Straight Lines Exercise Ex. 23.15

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Chapter 23 - The Straight Lines Exercise Ex. 23.16

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Find the ratio in which the line 3x + 4y + 2 = 0 divides the distance between the lines 3x + 4y + 5 = 0 and 3x + 4y - 5 = 0.

Solution 6

  

Chapter 23 - The Straight Lines Exercise Ex. 23.17

Question 1

Deduce the condition for these lines to form a rhombus.

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Chapter 23 - The Straight Lines Exercise Ex. 23.18

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Consider the following figure:

T h e space e q u a t i o n space o f space A B space i s
y minus 2 equals 1 fifth open parentheses x minus 1 close parentheses
rightwards double arrow 5 y minus 10 equals x minus 1
rightwards double arrow x minus 5 y plus 9 equals 0
A n d space t h e space e q u a t i o n space o f space B C space i s
y minus 8 equals minus 5 open parentheses x minus 5 close parentheses
rightwards double arrow y minus 8 equals minus 5 x plus 25
rightwards double arrow 5 x plus y minus 33 equals 0

Chapter 23 - The Straight Lines Exercise Ex. 23.19

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

rightwards double arrow fraction numerator 6 x minus 21 y plus 33 plus 7 x plus 21 y minus 56 over denominator 3 end fraction equals 0
rightwards double arrow 6 x minus 21 y plus 33 plus 7 x plus 21 y minus 56 equals 0
rightwards double arrow 13 x minus 23 equals 0
rightwards double arrow 13 x equals 23

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11


 

 

Chapter 23 - The Straight Lines Exercise Ex. 23VSAQ

Question 1

Solution 1

Question 2


Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9


Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13


Question 14


Solution 14

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