Question

Consider the parametric equationsx=cost -sint/√2y = cos t + sin t0 < t < 2pia. Eliminate the parameter to find a Cartesian equation for the parametric curve. Hint: Multiply by , then square both equations and add them together. b. Sketch the parametric curve, indicating with arrows the direction in which the curve is traced.

Answer 1

(a)
`x^2+2y^2=2`

(b)In the attached diagram

Explanation:

`x=cost -sint\\y√(2) =cost +sint, 0<t<2\pi`

Step 1: Multiply both equations by t

`xt=t(cost -sint)\\ty√(2) =t(cost +sint)`

Step 2:We square both equations

`(xt)^2=t^2(cost -sint)^2\\(ty)^2(√(2))^2 =t^2(cost +sint)^2`

Step 3: Adding the two equations

`(xt)^2+(ty)^2(√(2))^2=t^2(cost -sint)^2+t^2(cost +sint)^2\\t^2(x^2+2y^2)=t^2((cost -sint)^2+(cost +sint)^2)\\x^2+2y^2=(cost -sint)^2+(cost +sint)^2\\(cost -sint)^2+(cost +sint)^2=2\\x^2+2y^2=2`