Question

A particle is subjected to mutually perpendicular SHM’s such that its x and y coordinates are given by: x = 2 sin ωt and y = 2 sin [ωt + (π/4)] The path of the particle will be

• A. An ellipse
• B. A straight line
• C. A parabola
• D. A circle

Answer 1

Answer: (A)

An ellipse

Answer 2

y = 2 [ sin ωt . cos $\frac { \pi } { 4 }$ + cos ωt. sin $\frac { \pi } { 4 }$]

= $\frac { 2 } { \sqrt { 2 } }$[sin ωt + cos ωt]

=$\sqrt { 2 }$ [ + $\sqrt { 1 - \frac { x ^ { 2 } } { 4 } }$ ]

⇒ $\frac { y } { \sqrt { 2 } } = \frac { x } { 2 } + \frac { \sqrt { 4 - x ^ { 2 } } } { 2 }$

(y$\sqrt { 2 }$ − x)² = 4 – x²

or, 2x² + 2y² − 2 $\sqrt { 2 }$ xy – 4 = 0

or, x² + y² − $\sqrt { 2 }$ xy – 2 = 0

Which is the equation of an ellipse (h² < ab)