Question

Question #f1260

Answer 1

`y=+-5/4(16-x^2)^0.5`

Explanation:

`x=4costheta` & `y=5sintheta`
Square both sides
`x^2=16cos^2theta` & `y^2=25sin^2theta`
Notice that `sin^2theta=1-cos^2theta`
`y^2=25(1-cos^2theta)->` Equation 1

Substitute `cos^2theta=x^2/16` into Equation 1.

`y^2=25(1-x^2/16)`
Rearrange this to get:
`y=+-5/4(16-x^2)^0.5`

From your domain `-pi/2<=theta<=pi/2`

We find that x ranges from `{x;0<=x,+4}`
and y ranges from `{y;-5<=y<=5}`.
The fact that `x=4costheta` & `y=5sintheta` takes the form of a parametric curve (oval shape), we can draw an oval that is restricted by the given conditions.
graph{5/4(16-x^2)^0.5 [0, 4, -5, 5.5]}
The graph also reflects on the x-axis.
graph{-5/4(16-x^2)^0.5 [0, 4, -5, 5.5]}
Put the 2 graphs together.

The arrow should be pointing clockwise.