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The number of normal (s) of a rectangula...

The number of normal (s) of a rectangular hyperbola which can touch its conjugate is equal to

A

0

B

2

C

4

D

8

Text Solution

Verified by Experts

The correct Answer is:
C
Normal to hyperbola `xy = c^(2)` at `(ct, (c )/(t))` is `y - (c )/(t) = t^(2) (x-ct)` Solving with `xy =- c^(2)`, we get
`rArr x {(c )/(t) + t^(2) (x-ct)} +c^(2) =0`
`rArr t^(2) x^(2) + ((c )/(t)-ct^(3)) x +c^(2) =0`
Since line touches the curve, above equation has equal roots
`:. Delta = 0, ((1)/(t)-t^(3))^(2) - 4t^(2) =0`
`rArr (1-t^(4))^(2) - 4t^(4) =0`
`rArr t^(2) = (2=- sqrt(8))/(2)` or `(-2 +- sqrt(8))/(2)`
`rArr t^(2) =1 + sqrt(2)` or `-1 + sqrt(2)`
Thus four such values of t are possible
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