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