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Question

Show that the curves xy=a2 and x2+y2=2a2 touch each other.

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Solution

Given equation of the curves xy=a2(1) and x2+y2=2a2(2)

From (1)

y=a2x(3)

From (2)

x2+y2=2a2

x2+(a2x)2=2a2

x2+a4x2=2a2

x42a2x2+a4=0

(x2a2)2=0 ((ab)2=a22ab+b2)

x2a2=0

x=±a

y=a2x=±a

So point of intersection of both the curves is (a,a) and (a,a)

From (3)

dydx=a2x2

From (2)

2x+2ydydx=0dydx=xy

Slope of the tangent to curve xy=a2 at (a,a) is m1=dydx(a,a)=a2a2=1

Slope of the tangent to curve x2+y2=2a2 at (a,a) is m2=dydx(a,a)=aa=1

As we know that

If the angle between two lines with slopes m1,m2 is θ then tanθ=m1m21+m1m2

Let the angle between the tangents be θ

tanθ=1+11+(1)(1)=0

θ=0

Angle between the tangents is zero

So both tangents represents same line

Hence the curves xy=a2 and x2+y2=2a2 touch each other

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