Show that the curves and touch each other.

If the two curve touch each other then the tangent at their intersecting point formed a angle of 0.

We have to find the intersecting point of these two curves.

xy = a² and x² + y² = 2a²

⇒

⇒ x⁴ – 2a²x² + a⁴ = 0

⇒ (x² – a²) = 0

⇒ x = +a and -a

At x = a, y = a

At x = -a, y = -a

m₁ at (a, a) = -1

m₁ at (-a, -a) = -1

m₂ at (a, a) = -1

m₂ at (-a, -a) = -1

At (a, a)

⇒ θ = 0

At (-a, -a)

⇒ θ = 0

So, we can say that two curves touch each other because the angle between two tangent at their intersecting point is equal to 0.