Question

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

Answer 1

Given circles are xy = 4   .....(i)

And x² + y² = 8   .....(ii)

Differentiating equation (i) w.r.t., x

`x * "dy"/"dx" + y * 1` = 0

⇒ `"dy"/"dx" = - y/x`

⇒ m₁ = `- y/x`  .....(iii)

Where, m₁ is the slope of the tangent to the curve.

Differentiating equation (ii) w.r.t. x

`2x + 2y * "dy"/"dx"` = 0

⇒ `"dy"/"dx" = - x/y`

⇒ m₂ = `- x/y`

Where, m₂ is the slope of the tangent to the circle.

To find the point of contact of the two circles

m₁ = m₂

⇒ `- y/x = - x/y`

⇒  x² = y²

Putting the value of y² in equation (ii)

x² + x² = 8

⇒ 2x² = 8

⇒ x² = 4

∴ x = ± 2

∵ x2 = y2

⇒ y = ± 2

∴ The point of contact of the two circles are (2, 2) and (– 2, 2).