Question

If θ is a parameter, find the equation of the locus of a moving point, whose coordinates are x = a cos³θ, y = a sin³θ

Answer 1

The given moving points is (a cos³θ, a sin³θ)

x = a cos³θ

y = a sin³θ

`x/"a"` = cos³θ

`y/"a"` = sin³θ

cos θ = `(x/"a")^(1/3)`

sin θ = `(y/"a")^(1/3)`

cos²θ + sin²θ = `[(x/"a")^(1/3)]^2 + [(y/"a")^(1/3)]^2`

1 = `(x/"a")^(2/3) + (y/"a")^(2/3)`

1 = `(x^(2/3))/("a"^(2/3)) + (y^(2/3))/("a"^(2/3))`

`x^(2/3) + y^(2/3) = "a"^(2/3)` which is the required locus.