Question

Show that the two curves x² – y² = r² and xy = c² where c, r are constants, cut orthogonally

Answer 1

Given curves x² – y² = r²  ......(1)

xy = c²   .......(2)

 Let (x₁, y₁) be the point of intersection of the given curves.

(1) ⇒ x² – y² = r²

Differentiating w.r.t ‘x’

`2x - 2y  ("d"x)/("d"y)` = 0

`("d"x)/("d"y) = x/y`

Now `(("d"x)/("d"y))_(((x_1, y_1)))` = m₁

= `x_1/y_1`

(2) ⇒ xy = c²

Differentiating w.r.t ‘x’

`x ("d"y)/("d"x) + y*1` = 0

`("d"y)/("d"x) = - y/x`

 `(("d"x)/("d"y))_(((x_1, y_1)))` = m₂

= `- y_1/x_1`

Now, `"m"_1 xx "m"_2 = x_1/y_1  xx (- y_1/x_1) = - 1`

Hence, the given curves cut orthogonally.