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Q.

Let the line x–8y+K = 0, K I meets the rectangular hyperbola xy = 1 at the points whose abscissae are integers then the number of such lines is

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a

4

b

5

c

8

d

10

answer is A.

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Detailed Solution

Given hyperbola xy=1                             y=1x let P(x, y) be the point where the normal x-8y+K=0 meets the hyperbola x-81x+K=0 K=8x-x where KI Since KI then x should be x=±1, ±2, ±4, ±8 If x=1 then K=7,       If x=-1 then K=-7 If x=2 then K=2,       If x=-2 then K=-2 If x=4 then K=-2,   If x=-4 then K=2 If x=8 then K=-7,   If x=-8 then K=7 Here K=-7, -2, 2, 7    K has four values  four lines exist

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Let the line x–8y+K = 0, K∈ I meets the rectangular hyperbola xy = 1 at the points whose abscissae are integers then the number of such lines is