JEE Advance - Mathematics (1998 - No. 46)
The angle between a pair of tangents drawn from a point $$P$$ to the parabola $${y^2} = 4ax$$ is $${45^ \circ }$$. Show that the locus of the point $$P$$ is a hyperbola.
The equation of the pair of tangents from $$(h, k)$$ to the parabola $${y^2} = 4ax$$ is $${(y^2 - 4ax)(k^2 - 4ah) = {[ky - 2a(x + h)]}^2}$$.
The angle between the tangents is given by $${ an heta = left| { rac{{2sqrt {{k^2} - 4a(h - a)} }}{{h + a}}}
ight|}$$.
Given that the angle between the tangents is $$45^\circ$$, we have $${ an 45^\circ = 1}$$.
Therefore, $${\left| { rac{{2sqrt {{k^2} - 4a(h - a)} }}{{h + a}}}
ight| = 1}$$.
The locus of the point $${(h, k)}$$ is $${y^2 - x^2 - 6ax + a^2 = 0}$$, which is a hyperbola.
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