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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)2 + 4a2 t2 ) = a(1 + t2) = ST, so that
∠MPT = ∠PTS = ∠SPT ⇒ TP bisects ∠SPM.
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