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Analytical Geometry

Cool goIITian

Joined:	21 Apr 2009
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4 Mar 2010 11:36:53 IST

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Locus in parabola

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Show that the locus of the point of intersection of tangents to parobala y^2 =4ax, which intercept a constant length d on the directrix is (y^2-4ax)(X+a)^2=d^2x^2

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सुपर मान

Hot goIITian

Joined: 7 Nov 2009

Posts: 199

4 Mar 2010 12:25:37 IST

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refer to the figure

eq of tangent from at^2,2at : ty=x+at^2 put x= -a in the eq to get pt. A similarly find B

A comes out to be (-a,(-a+at^2)/t)

for B replace t by p

accordin to question AB=d

i.e. a(t^2 - p^2)=dtp .............................(1)

x=atp and y= a(t+p)

we have to eliminate t and p to get the locus of C

in eq. (1) put tp=x/a t+p=y/a and t-p=((y/a)^2 - 4x/a)^1/2

solve and u will get the locus

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