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Consider a hyperbola xy = 4 and a line y = 2x = 4. O is the centre of hyperbola. Tangent at any point P of hyperbola intersect the coordinate axes at A and B. Let the given line intersects the x-axis at R. if a line through R. intersect the hyperbolas at S and T, then minimum value of RS xx RT is |
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Answer» 2 `:. A = (4t,0),B = (0,4//t)` Locus of circumcentre of triangle is `xy = 16` Its eccentricity is `sqrt(2)` SHORTEST distance exist along the common NORMAL. `:. t^(2) = 1//2` or `t = 1//sqrt(2)` `:.` Foot of the perpendicular is `C (sqrt(2),2sqrt(2))` `:.` Shortest distance is distance of C from the given line which is `(4(sqrt(2)-1))/(sqrt(5))` Given line intersect the x-axis at `R(2,0)` Any point on this line at distance 'r' from R is `(2+r cos theta, r SIN theta)` If this point lies on hyperbola, then we have `(2+r cos theta) (r sin theta) =4` Product of roots of above quadratie in 'r' is `r_(1)r_(2) = 8//|sin 2 theta|`, which has minimum value 8 `:.` Minimum value of `RS xx RT` is 8 |
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