Find the perpendicular distance from the origin to the line joining the points
The points are ( cos θ , sin θ ) and ( cos ϕ , sin ϕ ) .
The formula for the equation of line passing through the points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by,
( y − y 1 ) = y 2 − y 1 x 2 − x 1 ⋅ ( x − x 1 ) (1)
Substitute the values of ( x 1 , y 1 ) and ( x 2 , y 2 ) as ( cos θ , sin θ ) and ( cos ϕ , sin ϕ ) respectively in equation (1).
( y − sin θ ) = sin ϕ − sin θ cos ϕ − cos θ ⋅ ( x − cos θ ) ( y − sin θ ) ⋅ ( cos ϕ − cos θ ) = ( sin ϕ − sin θ ) ⋅ ( x − cos θ ) y ⋅ ( cos ϕ − cos θ ) − sin θ ( cos ϕ − cos θ ) = x ⋅ ( sin ϕ − sin θ ) − cos θ ⋅ ( sin ϕ − sin θ ) − x ⋅ ( sin ϕ − sin θ ) + y ⋅ ( cos ϕ − cos θ ) = sin θ ( cos ϕ − cos θ ) − cos θ ⋅ ( sin ϕ − sin θ )
Further simplify the above expression.
x ⋅ ( sin θ − sin ϕ ) + y ⋅ ( cos ϕ − cos θ ) = sin θ ⋅ cos ϕ − sin θ ⋅ cos θ − cos θ ⋅ sin ϕ − cos θ ⋅ sin θ x ⋅ ( sin θ − sin ϕ ) + y ⋅ ( cos ϕ − cos θ ) = sin ( θ − ϕ ) x ⋅ ( sin θ − sin ϕ ) + y ⋅ ( cos ϕ − cos θ ) + sin ( ϕ − θ ) = 0
Compare the above equation with the general form of equation A x + B y + C = 0 .
A = ( sin θ − sin ϕ ) , B = ( cos ϕ − cos θ ) and C = sin ( ϕ − θ ) (2)
The formula for the perpendicular distance of the line A x + B y + C = 0 from a point ( x 1 , y 1 ) is given by,
d = | A x 1 + B y 1 + C | A 2 + B 2 (3)
Substitute the values of ( x 1 , y 1 ) as ( 0 , 0 ) , and the values of A, B and C from equation (2) in equation (3).
d = ( sin θ − sin ϕ ) ⋅ 0 + ( cos ϕ − cos θ ) ⋅ 0 + sin ( ϕ − θ ) ( sin θ − sin ϕ ) 2 + ( cos ϕ − cos θ ) 2 = | sin ( ϕ − θ ) | sin 2 θ + sin 2 ϕ − 2 sin θ sin ϕ + cos 2 ϕ + cos 2 θ − 2 cos ϕ cos θ = | sin ( ϕ − θ ) | ( sin 2 θ + cos 2 θ ) + ( sin 2 ϕ + cos 2 ϕ ) − 2 ( sin θ sin ϕ + cos ϕ cos θ ) = | sin ( ϕ − θ ) | 1 + 1 − 2 ( sin θ sin ϕ + cos ϕ cos θ )
Further simplify the above expression.
d = | sin ( ϕ − θ ) | 2 − 2 ( cos ( ϕ − θ ) ) = | sin ( ϕ − θ ) | 2 ( 1 − cos ( ϕ − θ ) ) = | sin ( ϕ − θ ) | 2 ( 2 sin 2 ( ϕ − θ 2 ) ) = | sin ( ϕ − θ ) | 4 ( sin 2 ( ϕ − θ 2 ) )
Further simplify the above expression.
d = | sin ( ϕ − θ ) | 2 sin ( ϕ − θ 2 )
Thus, the perpendicular distance from the origin to the line joining the points ( cos θ , sin θ ) and ( cos ϕ , sin ϕ ) is | sin ( ϕ − θ ) | 2 sin ( ϕ − θ 2 ) .
The points are
The formula for the equation of line passing through the points
Substitute the values of
Further simplify the above expression.
Compare the above equation with the general form of equation
The formula for the perpendicular distance of the line
Substitute the values of
Further simplify the above expression.
Further simplify the above expression.
Thus, the perpendicular distance from the origin to the line joining the points
