10.x= a (cos θ + θ sin θ), ya (sin θ-θ cos θ)
It is given that x and y are parametrically connected by the equations,
x = a ( cos θ + θ sin θ ) (1)
And,
y = a ( sin θ − θ cos θ ) (2)
Differentiate equation (2) with respect to θ .
d y d θ = a d ( cos θ + θ sin θ ) d θ d y d θ = a ( cos θ − cos θ + θ sin θ ) d y d θ = a ( θ sin θ ) d y d θ = a θ sin θ
Differentiate equation (1) with respect to θ .
d x d θ = a d ( cos θ + θ s i n θ ) d θ d x d θ = a ( − sin θ + sin θ + θ cos θ ) d x d θ = a ( θ cos θ )
We know that,
d y d x = d y d θ d x d θ
Substitute the value of d y d θ and d x d θ .
d y d x = a ( θ sin θ ) a ( θ cos θ ) d y d x = sin θ cos θ d y d x = tan θ
Thus, the solution is d y d x = tan θ .
It is given that
And,
Differentiate equation (2) with respect to
Differentiate equation (1) with respect to
We know that,
Substitute the value of
Thus, the solution is
