Question

The Cartesian equation of the curve given by \[ x = 6 \cos \theta, \quad y = 6 \sin \theta \] is

• A. \( x^2 + y^2 = 36 \)
• B. \( x^2 + y^2 = 5 \)
• C. \( x^2 + y^2 = 25 \)
• D. \( x^2 + y^2 = 6 \)

Hint:

For parametric equations involving trigonometric functions, use the Pythagorean identity to eliminate the parameter and find the Cartesian equation.

Answer 1

The Correct Option is A

Step 1: Use the parametric equations.
We are given the parametric equations \( x = 6 \cos \theta \) and \( y = 6 \sin \theta \). To eliminate the parameter \( \theta \), we use the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \).
Step 2: Square and add the equations.
Squaring both \( x \) and \( y \), we get: \[ x^2 = 36 \cos^2 \theta, \quad y^2 = 36 \sin^2 \theta. \] Adding these two equations, we get: \[ x^2 + y^2 = 36 (\cos^2 \theta + \sin^2 \theta) = 36 \times 1 = 36. \]
Step 3: Conclusion.
Thus, the Cartesian equation of the curve is \( x^2 + y^2 = 36 \), which corresponds to option (A).