To find the equation of an ellipse that circumscribes another ellipse, we need to start by identifying the parameters of the given ellipse.
The equation of the ellipse given is:
x² + 4y² = 4
This can be rewritten in standard form by dividing the entire equation by 4:
\( \frac{x²}{4} + \frac{y²}{1} = 1 \)
From this form, we can identify:
- Major semi-axis (a) = 2 (since \( a² = 4 \))
- Minor semi-axis (b) = 1 (since \( b² = 1 \))
This ellipse has its major axis along the x-axis. The center of this ellipse is at the origin (0,0).
To find the circumscribing ellipse, we can scale the axes by a factor. Since the original ellipse has axes lengths of 2 and 1, we can choose to scale these lengths. A common choice for a circumscribing ellipse is to scale by a factor greater than 1. Let’s use a factor of 2 for simplicity.
This would give us:
- New major semi-axis = \( 2 * 2 = 4 \)
- New minor semi-axis = \( 2 * 1 = 2 \)
Now, writing this in standard form, we get:
\( \frac{x²}{16} + \frac{y²}{4} = 1 \)
Therefore, the equation of the ellipse that circumscribes the given ellipse is:
x²/16 + y²/4 = 1