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
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)
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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