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Question

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

A
8x2y2+2xy=2
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B
16x2+y2+10xy=2
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C
8x2+y2+2xy=2
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D
16x2y2+10xy=2
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Solution

The correct option is B 16x2+y2+10xy=2
Let two points on the hyperbola A(t1,1t1) and B(t2,1t2)
slope of AB=4
t1t2=14(1)
Now let P(h,k) on the line AB

Case 1: AP:PB=1:2
h=2t1+t23,k=t1+2t23t1t2t1=8h+k4,t2=(k+2h)2
using (1)
(8h+k)(k+2h)=2
Hence locus will be
16x2+y2+10xy=2

Case 2: AP:PB=2:1
h=t1+2t23,k=2t1+t23t1t2t1=(2h+k)2,t2=8h+k4
using (1)
(8h+k)(k+2h)=2
Hence locus will be
16x2+y2+10xy=2

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