Question

The parametric form of the ellipse \( 4(x + 1)^2 + (y - 1)^2 = 4 \) is:

• A. \( x = \cos \theta - 1, \, y = 2\sin \theta - 1 \)
• B. \( x = 2\cos \theta - 1, \, y = \sin \theta + 1 \)
• C. \( x = \cos \theta - 1, \, y = 2\sin \theta + 1 \)
• D. \( x = \cos \theta + 1, \, y = 2\sin \theta + 1 \)
• E. \( x = \cos \theta + 1, \, y = 2\sin \theta - 1 \)

Hint:

In parametric forms, the constant terms are the coordinates of the center, and the coefficients of the trig functions are the lengths of the semi-axes.

Answer 1

The Correct Option is C

Understanding the Concept: For an ellipse in the form \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), the parametric equations are \( x = h + a\cos\theta \) and \( y = k + b\sin\theta \).

Step 1: Convert to standard form.
Divide the equation \( 4(x + 1)^2 + (y - 1)^2 = 4 \) by 4: \[ \frac{(x + 1)^2}{1} + \frac{(y - 1)^2}{4} = 1 \] \[ \frac{(x - (-1))^2}{1^2} + \frac{(y - 1)^2}{2^2} = 1 \]

Step 2: Identify parameters.
Center \( (h, k) = (-1, 1) \). Semi-axes \( a = 1 \) and \( b = 2 \).

Step 3: Write parametric equations.
\[ x = -1 + 1\cos\theta = \cos\theta - 1 \] \[ y = 1 + 2\sin\theta = 2\sin\theta + 1 \]