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.

Options

A. `16x^(2)+10xy+y^(2)=2`
B. `16x^(2)-10xy+y^(2)=2`
C. `16x^(2)+10xy+y^(2)=4`
D. none of these

Explanation

Let `P(h,k)` be the point dividing `AB` internally in the ratio `1 :2`. Then,
`3h=2x_(1)+x_(2)` and `3k=2y_(1)+y_(2)`

The equation of `AB` is `y-k=4(x-h)`……`(i)`
The abscissae of the points of intersection of `(i)` and the hyperbola `xy=1` are the roots of the equation
`4x^(2)-(4h-k)x-1=0`
`:.x_(1)+x_(2)=(4h-k)/(4)` and `x_(1)x_(2)=-(1)/(4)`
On eliminating `x_(1)`, `x_(2)` , we get
`16h^(2)+10hk+k^(2)=2`
Hence,the locus of `(h,k)` is `16x^(2)+10xy+y^(2)-2`