Question

Parametric equations of the circle \( 2x^2 + 2y^2 = 9 \) are:

• A. \( x = \frac{3}{2} \cos\theta, \quad y = \frac{3}{2} \sin\theta \)
• B. \( x = \frac{3}{\sqrt{2}} \cos\theta, \quad y = 3 \sin\theta \)
• C. \( x = \frac{3}{\sqrt{2}} \sin\theta, \quad y = \frac{3}{\sqrt{2}} \cos\theta \)
• D. \( x = 3 \sin\theta, \quad y = \frac{3}{2} \cos\theta \)

Hint:

For a circle centered at \( (0,0) \) with radius \( r \), the parametric equations are: \[ x = r \cos\theta, \quad y = r \sin\theta. \] If a transformation occurs, check for sine-cosine interchanges.

Answer 1

The Correct Option is C

Step 1: Convert the Given Equation to Standard Form The given equation of the circle is: \[ 2x^2 + 2y^2 = 9. \] Dividing throughout by 2: \[ x^2 + y^2 = \frac{9}{2}. \] This represents a circle centered at \( (0,0) \) with radius: \[ r = \sqrt{\frac{9}{2}} = \frac{3}{\sqrt{2}}. \] Step 2: Standard Parametric Equations of a Circle The standard parametric form of a circle \( x^2 + y^2 = r^2 \) is: \[ x = r \cos\theta, \quad y = r \sin\theta. \] Substituting \( r = \frac{3}{\sqrt{2}} \): \[ x = \frac{3}{\sqrt{2}} \cos\theta, \quad y = \frac{3}{\sqrt{2}} \sin\theta. \] However, if we switch sine and cosine: \[ x = \frac{3}{\sqrt{2}} \sin\theta, \quad y = \frac{3}{\sqrt{2}} \cos\theta. \] This matches option (C).
Step 3: Conclusion
Thus, the correct answer is: \[ \mathbf{x = \frac{3}{\sqrt{2}} \sin\theta, \quad y = \frac{3}{\sqrt{2}} \cos\theta}. \]