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Show that the tangent at any point P of the parabola bisecys angle between focal chord to P and perpendicular from P to directrix.

Show that the tangent at any point P of the parabola bisecys angle between focal chord to P and perpendicular from P to directrix.

Grade:12th pass

1 Answers

Arun
25763 Points
3 years ago
 
  • The tangent at any point P on a parabola bisects the angle between the focal chord through P and the perpendicular from P on the directrix.

?The tangent at P (at2, 2at) is ty = x + at2.

It meets the x-axis at T(–at2, 0).

Hence, from the figure given above ST = SA + AT = a (1 + t2). 

Also, SP = √(a2(1 + t2)+ 4a2 t2 ) = a(1 + t2) = ST, so that 

∠MPT = ∠PTS = ∠SPT ⇒ TP bisects ∠SPM.

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