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Question

# A particle is subjected to two mutually perpendicular simple harmonic motions such that its x and y coordinates are given by x=2sinωt ; y=2sin(ωt+π4). The path of the particle will be:

A
an ellipse
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B
a straight line
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C
a parabola
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D
a circle
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Solution

## The correct option is A an ellipsex=2sinωt;y=2sin(ωt+π4)⇒sinωt=x2 .....(I) ⇒cosωt=√1−sin2ωt=√1−(x2)2=√1−x24 .....(II)y=2sin(ωt+π4)⇒y=2(sinωt+cosωt)√2=√2(sinωt+cosωt)using the values of from (I) and (II), we havey=√2(x2+√1−x24)⇒y√2−x2=√1−x24⇒y22+x24−xy√2=1−x24⇒x22+y22−xy√2−1=0⇒x2+y2−√2xy−2=0which is an equation of an oblique ellipse.

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