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

## The correct option is C 16x2+y2+10xy=2Let 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+2t23t1t2⇒t1=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+t23t1t2⇒t1=−(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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