Both the x and y coordinates of a point execute simple harmonic motion. The frequencies are the same but the amplitudes are different. The resulting orbit might be:

the orbit resulting an ELIPSE

Explanation:

Harmonic motion is described by the expression

          x = A cos (wt +Ф₁)

In this exercise it is established that in the y axis there is also a harmonic movement with the same frequency, so its equation is

        y = B cos (wt + Ф₂)

the combined motion of the two bodies can be found using the Pythagorean theorem

        R² = x² + y²

        R² = [A² cos² (wt + Ф₁) + B² cos² (wt + Ф₂)]

to simplify we can assume that the phase in the two movements are equal

        R = √(A² + B²) cos (wt + Ф)

        If the two amplitudes are equal we have a circular motion, if the two amplitudes are different in elliptical motion, the amplitudes of the two motions are

circular    R² = (A² + A²)

elliptical   R² = (A² + B²)

We see from the last expression that the broadly the two axes is different, so the amplitude is an ellipse.

By which the orbit resulting an ELIPSE

the first step is making objective observations.

they have to be based on a certain event and can be confirmed by others if it is right or not.