Find the condition for the following set of curves to interest orthogonally.

and

Given:

Curves 1 ...(1)

& 1 ...(2)

First curve is 1

Differentiating above w.r.t x,

⇒ . = 0

⇒ .

⇒

⇒

⇒ m₁ ...(3)

Second curve is 1

Differentiating above w.r.t x,

⇒ . = 0

⇒ .

⇒

⇒

⇒ m₁ ...(4)

When m₁ & m₂ =

Since ,two curves intersect orthogonally,

⇒ × = – 1

⇒ × = – 1

⇒ ...(5)

Now equation (1) – (2) gives

⇒ x²() – y²() = 0

⇒ x²() = y²()

⇒

⇒

⇒

Substituting from equation (5),we get

⇒

⇒ – 1

⇒ ( – 1)(A² – a²) = (B² – b²)

⇒ a² – A² = B² – b²

⇒ a² + b² = B² + A²