if x=2a, y=2b show that xy=2a+b
Answers
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Answer:
Given equation of the curves xy=a
2
⋯(1) and x
2
+y
2
=2a
2
⋯(2)
From (1)
y=
x
a
2
⋯(3)
From (2)
⟹x
2
+y
2
=2a
2
⟹x
2
+(
x
a
2
)
2
=2a
2
⟹x
2
+
x
2
a
4
=2a
2
⟹x
4
−2a
2
x
2
+a
4
=0
⟹(x
2
−a
2
)
2
=0 (∵(a−b)
2
=a
2
−2ab+b
2
)
⟹x
2
−a
2
=0
⟹x=±a
y=
x
a
2
=±a
So point of intersection of both the curves is (a,a) and (−a,−a)
From (3)
dx
dy
=−
x
2
a
2
From (2)
2x+2y
dx
dy
=0⟹
dx
dy
=−
y
x
Slope of the tangent to curve xy=a
2
at (a,a) is m
1
=
dx
dy
∣
∣
∣
∣
∣
(a,a)
=−
a
2
a
2
=−1
Slope of the tangent to curve x
2
+y
2
=2a
2
at (a,a) is m
2
=
dx
dy
∣
∣
∣
∣
∣
(a,a)
=−
a
a
=−1
As we know that
If the angle between two lines with slopes m
1
,m
2
is θ then tanθ=
1+m
1
m
2
m
1
−m
2
Let the angle between the tangents be θ
⟹tanθ=
1+(−1)(−1)
−1+1
=0
⟹θ=0
Angle between the tangents is zero
So both tangents represents same line
Hence the curves xy=a
2
and x
2
+y
2
=2a
2
touch each other