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Analytical Geometry

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5 Jun 2009 23:45:16 IST

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Straight lines A variable line through the point of intersection of the lines x/a+y/b=1 and x/b+y

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Straight linesA variable line through the point of intersection of the lines x/a+y/b=1 and x/b+y/a=1 meets the coordinate axes in A and B. Show that the locus of the midpoint of AB is the curve 2xy(a+b)=ab(x+y).

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Akshay

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Joined: 27 Dec 2007

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8 Jun 2009 17:05:41 IST

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eqn of a line passing through the intersection of the given lines=(x/a+y/b-1) +k(x/b+y/a-1)=0

now, recast this eqn in intercept form

x(1/a+k/b)+y(1/b+k/a)=k+1

x[(b+ak)/ab(k+1)] + y(a+bk)/ab(k+1)=1

so x intercept=ab(k+1)/(b+ak)=A

and y intercept=ab(k+1)/(a+bk)+B

so midpoint of AB is given by xcoordinate=l=ab(k+1)/2(b+ak)

m=ab(k+1)/2(a+bk)=y coordinate

eliminate k and obtain a relation between l and m

replace l and m by x and y respectively

Mirka

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Joined: 13 Aug 2008

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8 Jun 2009 18:04:54 IST

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Continuing from above ...

Proved.

Mirka

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Joined: 13 Aug 2008

Posts: 1313

8 Jun 2009 18:30:39 IST

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im sorry der's a typo above .. it should be 1st line dnominator for x is 2(ak+b) and for y its 2 in place of 'a' .... rest OK

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