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Class 11-commerce RD SHARMA Solutions Maths Chapter 23: The Straight Lines

The Straight Lines Exercise Ex. 23.1

Solution 1

Solution 2

Solution 3(i)

Solution 3(ii)

Solution 3(iii)

Solution 3(iv)

Solution 4

Solution 5(i)

Solution 5(ii)

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

The Straight Lines Exercise Ex. 23.2

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

The Straight Lines Exercise Ex. 23.3

Solution 1

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

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

The Straight Lines Exercise Ex. 23.4

Solution 1

Solution 2

Solution 5

Solution 6

Solution 7

Solution 8

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

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 3

Solution 4

The Straight Lines Exercise Ex. 23.5

Solution 1(i)

Solution 1(ii)

Solution 1(iii)

Solution 1(iv)

Solution 1(v)

Solution 1(vi)

Solution 2(i)

Solution 2(ii)

Solution 3

Solution 4

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

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

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

Solution 12

Solution 13

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

Solution 14

Solution 15

 

 

 

 

The Straight Lines Exercise Ex. 23.6

Solution 1

Solution 2

Solution 3

3

Solution 4


 

 

 

Solution 5

Solution 6

 

 

Solution 7

Solution 8

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

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

The Straight Lines Exercise Ex. 23.7

Solution 1(i)

Solution 1(ii)

Solution 1(iii)

Solution 1(iv)

Solution 2


 

 

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

The Straight Lines Exercise Ex. 23.8

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

The Straight Lines Exercise Ex. 23.9

Solution 1

Solution 2(i)

Solution 2(ii)

Solution 2(iii)

Solution 2(iv)

Solution 2(v)

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

The Straight Lines Exercise Ex. 23.10

Solution 1(i)

Solution 1(ii)

Solution 1(iii)

Solution 2(i)

Solution 2(ii)

Solution 3(i)

Solution 3(ii)

Solution 3(iii)

Solution 4

Solution 5

Solution 6(i)

Solution 6(ii)

Solution 6(iii)

Solution 7

Solution 8

Solution 9

 

Solution 10

Solution 11

 

Solution 12

Solution 13

Solution 14

 

Solution 15

Solution 16


 

 

Solution 17

 

The Straight Lines Exercise Ex. 23.11

Solution 1(i)

Solution 1(ii)

Solution 1(iii)

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

The Straight Lines Exercise Ex. 23.12

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

 

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25

Solution 26


Solution 27


 

 

The Straight Lines Exercise Ex. 23.13

Solution 1(i)

Solution 1(ii)

Solution 1(iii)

Solution 1(iv)

Solution 1(v)

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10


 

 

The Straight Lines Exercise Ex. 23.14

Solution 1

Solution 2

Solution 3

The Straight Lines Exercise Ex. 23.15

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 15

Solution 12

Solution 13

Solution 14

The Straight Lines Exercise Ex. 23.16

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

 

The Straight Lines Exercise Ex. 23.17

Solution 1

Solution 2

Solution 3

The Straight Lines Exercise Ex. 23.18

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

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

The Straight Lines Exercise Ex. 23.19

Solution 1

Solution 2

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

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11


 

 

The Straight Lines Exercise Ex. 23VSAQ

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14


Solution 15

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