Prove that the curves xy = 4 and x²+y² = 8 touch each other.

Given: two curves x² + y² = 8 and xy = 4

To prove: the two curves touch each other

Explanation:

Now given x² + y² = 8

Differentiating this with respect to x, we get

Applying sum rule of differentiation, we get

Also given xy = 4

Differentiating this with respect to x, we get

Now applying the product rule of differentiation, we get

But the two curves touch each other, if

m₁ = m₂

Now substituting the values from equation (ii) and equation (ii), we get

⇒ y² = x²

⇒ x = y…….(iii)

Now substituting x = y in x² + y² = 8, we get

y² + y² = 8

⇒ 2 y² = 8

⇒ y² = 4

⇒ y = ±2

When y = 2,

xy = 4 becomes

x(2) = 4⇒ x = 2

when y = -2,

xy = 4 becomes

x(-2) = 4⇒ x = -2

Hence the point of intersection of the two curve is (2,2) and (-2, -2)

Substituting these points of intersection equation (i) and equation (ii), we get

For (2,2),

∴ m₁ = m₂

For (-2,-2),

∴ m₁ = m₂

Therefore, for both curves to touch each other, the slopes of both the curves should be same.

Hence the two given curves touch each other.

Hence proved