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 Ratio 1 : 2 Is

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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 ratio 1 : 2 is

Solution

Correct option is

16x2 + 10xy + y2 = 2

 

P(h, k) be any point on the variable line with slope = 4. Hence it equation is y – k = m(x – h)                                        …… (1)

m = 4, let this line (1) cut the hyperbola xy = 1            …… (2)

at A(x1, y1) and B(x2, y2). By assumption, P divides AB in the ratio 1 : 2 then 

Thus ⇒ 2x1 + x2 = 3h , 2y1 + y2 = 3k                             …... (3)

By (2), , put this in (1), we get 

⇒ 4x2 – (4h – k) x – 1 = 0                                               …… (4)

x1, x2 are roots of (4)                          …… (5)

                                                                         …… (6)

Solving (3) and (4), for x1 and x2, we get

       

Putting these in (6) 

Locus of (h, k) = 16x2 + 10xy – 2 = 0.

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