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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Solution

The correct option is A

Let P(h, k)

y - k = 4(x - h) --- (1)

Let it meets xy = 1 ----(2) at A $(x_{1}, y_{1}) a n d B (x_{2}, y_{2})$

$x_{1} + x_{2} = \frac{4 h - k}{4}, x_{1} x_{2} = - \frac{1}{4} A l s o \Rightarrow ∴ \frac{2 x_{1} + x_{2}}{3} = h \Rightarrow x_{1} = \frac{8 h + k}{4}, x_{2} = \frac{2 h + k}{2}$

$\Rightarrow 16 x^{2} + 10 x y + y^{2} = 2$

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