Question

A particle is projected over a traingle from one end of a horizontal base and grazing the vertex falls on the other end of the base. If alpha and beta be the base angles and theta the angle of projection, prove that tan theta = tan alpha + tan beta.

Answer 1

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Answer 2

Given:

A particle is thrown over a triangle from one end of a horizontal base and after grazing the vertex falls on the other end of the base.

α and β be the base angles.

Ф the angle of projection.

To Prove:

tan Ф = tan α + tan β

Solution:

Let the height of the triangle be h .

Let x be the distance of the foot of the perpendicular (from the top vertex to the base.) from the point of projection.

Then ,

tan α = y/x

tan β = y/R - x

tan α + tan β = y/x + y/R - x

tan α + tan β = yR/x(R-x) -- (a)

Now consider equation of trajectory,

y = x tanФ ( 1- x/R)

yR/(R - x) = x tan Ф

tan Ф =  yR/x(R-x) -- (b)

Therefore,

equating LHS of (a) and (b)

tan α + tan β = tan Ф

Thus proved that tan Ф =  tan α + tan β

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