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Question

Consider a hyperbola xy=4 and a line 2x+y=4. Let the given line intersect the xaxis at R. If a line through R intersects the hyperbola at S and T, then the minimum value of RS×RT=

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Solution

Given line intersect the xaxis at R(2,0)
Any point on the line which passes through R and at a distance of r units from R is
(2+rcosθ,rsinθ)

If this point lies on hyperbola xy=4, then we have (2+rcosθ)(rsinθ)=4
r2sinθcosθ+2rsinθ4=0
It is Quadratic in terms of r, whose roots are RS,RT
Product of roots of the above quadratic equation in r is r1r2=RS×RT=8|sin2θ|,
whose minimum value =8 (0|sin2θ|1)
Minimum value of RS×RT=8

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