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