Question

which of the following point lies on 2x-3y=6​

Answer 1

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The points (x, y) lies on the line 2x+3y=6. The smallest value of the quantity

x

2

+y

2

, is?

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ANSWER

It is given that point (x,y) lies on 2x+3y=6.

∴x=

2

6−3y

...(1) (rearranging)

Now,

x

2

+y

2

=

(

2

6−3y

)

2

+y

2

(substituting from (1))

∴

x

2

+y

2

=

(

4

36−36y+9y

2

)+y

2

∴

x

2

+y

2

=

4

36−36y+13y

2

Differentiating expression wrt y and equating to 0, we get

2

1

4

36−36y+13y

2

1

4

26y−36

=0

∴26y−36=0 (ensuring 36−36y+13y

2



=0 for obtained solution)

∴y=

13

18

Hence, minima of function

x

2

+y

2

occurs at y=

13

18

.

∴

x

2

+y

2

=

4

36−36×

13

18

+13(

13

18

)

2

∴

x

2

+y

2

=

13

9×13−9×18+9×9

∴

x

2

+y

2

=

13

36

∴

x

2

+y

2

=6

13

13

This is the required answer.