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Question

A variable straight line of slope 4 intersects the hyperbola xy=1 at two points. Find the locus of the point which divides the line segment between these two points in the ratio 1:2

A
16x2+y2+10xy=2
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B
16x2y210xy=2
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C
16x2y2+10xy=2
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D
None of these
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Solution

The correct option is A 16x2+y2+10xy=2
Let y=4x+c meets xy=1 at two points A and B.
A(t1,1t1),B(t2,1t2)
Therefore coordinates of P are ⎜ ⎜ ⎜2t1+t22+1,2.1t1+1.1t22+1⎟ ⎟ ⎟=(h,k) (say)
h=2t1+t23 and k=2t2+t13t1t2 ...(1)
Also (t1,1t1) and (t2,1t2) lies on y=4x+c
1t21t1t2t1=1t1t2=4t1t2=14 ...(2)
From equation (2)
t1=2h+k4 and t2=hk2 ...(3)
From equation (2) and (3)
(hk2)(2h+k4)=14(2h+k2)(8h+k4)=14(2h+k)(8h+k)=216h2+k2+10hk=2
Hence, required locus is 16x2+y2+10xy=2

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