Consider two simple harmonic motion along x and y- axis having same frequencies but different amplitudes as `x=A sin (omega t+varphi)` (along x axis) and `y= B sin omega t`( along y axis).
then show that `(x^(2))/(A^(2))+(y^(2))/(B^(2))-(2xy)/(AB) cos varphi = sin^(2) varphi` and also discuss the special cases when
`varphi=(pi)/(2) and A=B`
Note : when a particle is subjected to two simple harmonic motion at right angle to each other the particle may move along different paths.

Explanation

Note: When a particle is subjected to two simple harmonic motion at right angle to each other the particle may move along different paths. Such paths are called Lissajous figures.
(a) `y=(B)/(A)x`, equation is a straight line through origin with positive slope.
(b) `y=-(B)/(A)x`, equation is a straight line passing throug origin with negative slope.
(c ) `(x^(2))/(A^(2))+(y^(2))/(B^(2))=1`, equation is an ellipse whose center is origin.
(d) `x^(2)+y^(2)=A^(2)`, equation is a circle whose center is origin
(e) `(x^(2))/(A^(2))+(y^(2))/(B^(2))-(2xy)/(AB)(1)/(sqrt(2))=(1)/(2)`, equation is an ellipse (oblique ellipse which means tilted ellipse)