Answer
The parametric form of the answer is:
$$x=a\,cos\theta\\y=b\,sin\theta$$
Removing the parameter, we get the ellipse:
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$
Work Step by Step
Draw a line such that the angle between the line and the $x$-axis is $\theta$. Draw two line segments perpendicular to the $x$-axis with each going through points where the line we drew earlier intersects with each circle.
Looking at the drawing, we can see that the distance point $P$ is from the $x$-axis is equal to the height of the smaller triangle and the distance point $P$ is from the $y$-axis is equal to the width of the larger triangle.
Since the hypotenuse of the smaller triangle is $b$ and the hypotenuse of the larger triangle is $a$, we can represent all possible points $P$ with the equations:
$$x=a\,cos\theta\\y=b\,sin\theta$$
To remove the parameter, we can square each equation, getting:
$$x^2=a^2cos^2\theta\\y^2=b^2sin^2\theta$$
We can divide each equation by $a^2$ and $b^2$ respectively getting:
$$\frac{x^2}{a^2}=cos^2\theta\\\frac{y^2}{b^2}=sin^2\theta$$
Adding the two equations together, we get:
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=cos^2\theta+sin^2\theta$$
Using the Pythagorean trig identity, we get the ellipse:
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$
