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Analytical Geometry

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2 Oct 2007 22:42:14 IST

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A variable straight line of slope 4 intersects the hyperbola xy=1 at 2 points. Find the locus of the point which divides the line segments between these two points in the ratio 1:2

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Avinash Sharma

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Joined: 9 Mar 2007

Posts: 259

4 Nov 2007 16:36:34 IST

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Hello vkn128

The general equation of the line is y=mx+c here m is slope. Here m=4 given so the line is

y=4x+c ???????. (1)

Which cut the hyperbola ( xy = 1) at (x₁,y₁) and (x₂,y₂),so intersecting points are

x.( 4x+c) =1 or 4x² +cx -1 =0

or x₁ & x₂ are =[ {-c ± Ö (c² + 4c)}/ 8]

or x₁ =[ {-c + Ö (c² + 4c)}/ 8] and x₂ =[ {-c - Ö (c² + 4c)}/ 8] ????????. (ii)

Lets point (h,K) cut the line segment in 1:2 ration it mean

h = {2x₁ + x₂}/3

using (ii) we get h = {-3c+Ö (c² + 4c)}/24 ??????. (iii)

but (h,k) lies on the line (i) so

k=4h+c

or c = k - 4h ?????? (iv)

Now using (iii) & (iv) remove ?c? and you will get a relation in h and k replace h with x and k with y and you will get the desired locus.

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