Question

The area of the region, inside the ellipse \(x^2+4y^2=4\) and outside the region bounded by the curves \(y=x-1\) and \(y=1-x\), is:

• A. \(2\pi-1\)
• B. \(3(\pi-1)\)
• C. \(2(\pi-1)\)
• D. \(2\pi - \frac{1}{2}\)

Hint:

When a geometry problem's description seems ambiguous, look for clues in the answer options.
The format of the options can often reveal the intended interpretation. In this case, \(2\pi-2\) strongly implies subtracting an area of 2 from the ellipse's area, guiding you to find a simple geometric shape with area 2 that fits the description.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to find the area of a region defined by two conditions: it must be inside a given ellipse and outside a region defined by two intersecting lines. The wording "region bounded by the curves y=x-1 and y=1-x" is ambiguous. However, by looking at the options, we can infer the intended meaning. The answers are of the form (Area of Ellipse) - (Some Area). The answer key suggests the result is \(2\pi - 2\). This implies we need to subtract an area of 2 from the ellipse's area.
Step 2: Key Formula or Approach:
1. Area of Ellipse: The area of an ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) is given by \(\pi ab\).
2. Required Area: Required Area = Area(Ellipse) - Area(Region common to ellipse and the region to be excluded).
3. Interpreting the excluded region: The lines \(y=x-1\) and \(y=1-x\) along with \(y=-(x+1)\) and \(y=x+1\) form the boundary of a rhombus (a square rotated by 45 degrees) with vertices at (1,0), (0,1), (-1,0), and (0,-1). The equation for this rhombus is \(|x|+|y|=1\). The area of this rhombus is 2. It is highly likely that this is the intended excluded region, despite the poor wording.
Step 3: Detailed Explanation:
Part 1: Area of the Ellipse
The equation of the ellipse is \(x^2+4y^2=4\). Dividing by 4, we get the standard form: \[ \frac{x^2}{4} + \frac{y^2}{1} = 1 \] Here, \(a^2=4 \implies a=2\) and \(b^2=1 \implies b=1\). The area of the ellipse is \(A_{\text{ellipse}} = \pi ab = \pi(2)(1) = 2\pi\). Part 2: Area of the Excluded Region
As reasoned above, the form of the answer \(2\pi - 2\) suggests the area to be excluded is 2. Let's verify that the rhombus defined by \(|x|+|y|=1\) has an area of 2 and is reasonably described by the problem statement.
The vertices of this rhombus are (1,0), (0,1), (-1,0), and (0,-1). The lengths of the diagonals are \(d_1 = 1 - (-1) = 2\) and \(d_2 = 1 - (-1) = 2\).
The area of the rhombus is \(A_{\text{rhombus}} = \frac{1}{2} d_1 d_2 = \frac{1}{2} (2)(2) = 2\).
The lines bounding this rhombus are \(y=x-1\), \(y=1-x\), \(y=-x-1\), and \(y=x+1\). The question only mentions the first two, which is an ambiguity. However, this interpretation fits the provided correct answer.
The rhombus is entirely contained within the ellipse since its vertices lie on or inside the ellipse.
Part 3: Required Area
The required area is the area inside the ellipse but outside the rhombus. \[ A_{\text{required}} = A_{\text{ellipse}} - A_{\text{rhombus}} \] \[ A_{\text{required}} = 2\pi - 2 = 2(\pi-1) \] Step 4: Final Answer:
The area of the specified region is \(2(\pi-1)\).