Question

Find `"dy"/"dx"` if : x = a cos³θ, y = a sin³θ at θ = `pi/(3)`

Answer 1

a cos³θ, y = a sin³θ
Differentiating x and y w.r.t. θ, we get
`"dx"/"dθ" = a"d"/"dθ"(cosθ)^3`

= `a xx 3cos^2θ."d"/"dθ"(cosθ)`
= 3a cos²θ(– sinθ)
= – 3a cos²θ sinθ
and
`"dy"/"dθ" = a"d"/"dθ"(sinθ)^3`

= `a xx 3 sin^2θ."d"/"dθ"(sinθ)`
= 3a sin²θ cosθ
∴`"dy"/"dx" = (("dy"/"dθ"))/(("dx"/"dθ")`

= `(3a sin^2θ cosθ)/(-3a cos^2θ sinθ)`

= – tanθ

∴ `(dx/dy)_("at"  θ - pi/3)`

= `-tan  pi/(3)`
= `-sqrt(3)`