The plane region R=xyx2+y2≤4x24+y21≥1 is Ay-simpleBr-simpleCNone of the othersDx-simple

To determine the type of region R, we need to see which variable, x or y, is easier to express as a function of the other variable within the given inequalities.

Starting with the first inequality, we can rearrange it as y^2 ≤ 4 - x^2, which means y is bounded by the square root of the difference between 4 and x^2. This suggests that it may be easier to express y as a function of x, so we can rewrite the first inequality as:

−√(4−x^2) ≤ y ≤ √(4−x^2)

Now, looking at the second inequality, we can rearrange it as y^2 ≥ 1 − x^2/4, which means y is bounded below by the square root of 1 minus x^2/4. This suggests that it may be easier to express x as a function of y, so we can rewrite the second inequality as:

x^2 ≤ 4(1 − y^2)

or

|x| ≤ 2√(1−y^2)

Putting these together, we can see that the region R is bounded by the curves:

y = √(4 − x^2) and y = −√(4 − x^2) (from the first inequality)

and

x = 2√(1 − y^2) and x = −2√(1 − y^2) (from the second inequality)

This means that R is a r-simple region, which is answer choice B