Coordinate Geometry

[Tanish Desai]

In a Triangle ABC it is given that

tanAtan(B/2)=2

and the length of AB is fixed “a” unit.

Show that the locus of C is a Parabola with vertex A and focus B.

[Shirsendu Roy]

A(0,0),B(a,0) C(x,y)

tanAtanB/2=2 is a given equation

2tan(B/2)/{1-tan^{2}(B/2)}=tanB

or, 2{2/(y/x)}/[1-{2/(y/x)}^{2}]=(y-0)/(x-a) putting given equation where tanA=y/x

or, 4xy/(y^{2}-4x^{2})=(y-0)/(x-a)

or, y^2=4x(2x-a) which is required equation of parabola.

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