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Analytical Geometry

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4 Dec 2007 17:06:08 IST

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PARABOLA

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Two perpendicular straight lines through the focus of the parabola y^2=4ax meet its directrix in T & T' respectively. Show that the tangents to the parabola parallel to the perependicular lines intersect in the mid-point of TT'

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sowjanya gudipati

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Joined: 13 Oct 2007

Posts: 75

4 Dec 2007 19:13:10 IST

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in the parabola focus is S=(-a,0) two perpendicular lines from focus intersect the directrixat T and T1

let T=(-a,k1)and T1=(-a,k2)

slope of line joining S and T be m1=-k1/2a and slope of line joining S and T1 be m2=-k2/2a

know equation of tangent which is parallel to ST is y=(-k1/2a)x-2a²/k1 and equation of tangent parallel to ST2 is y=(-k2/2a)x-2a²/k2

know solve both the tangent equation for point of intersection which comes to be(-a,k1+k2/2) which is indeed the midpoint of T and T1

hey don't forget to use k1.k2=-4a² which is obtained by using the info the both lines are perpendicular i.e m1.m2=-1

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