IIT JEE , AIEEE Forum - Mathematics , Analytical Geometry

ds

If "T" be any point on the tangent at any point P on the parabola,& if TL be perpendicular to the focal radius SP & TN be prpendicular to the directrix,prove that SL=TN.Hence obtain a geometric construction for the pair of tangents drawn to the parabola from any point T.

magiclko

in the attachd figure, PQ is perpendicular to the directrix and RT is perpendicular to PQ.

thus the tangent will bisect the angle QPS, i.e.

AngleQPT = AngleSLT

also, PQ =PS (by definition of parabola)

and PT is common for traingles PRT and PTL...

thus the two triangles are congruent...giving PR=PL

now PQ = PS

=> PR + RQ = PL + LS

=> QR = SL

=> TN = SL

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