The area of the region, inside the ellipse x^2 + 4y^2 = 4$x^{2} + 4y^{2} = 4$ and outside the region bounded by the curves y = |x| - 1$y = |x| - 1$ and y = 1 - |x|$y = 1 - |x|$ , is:

### Related Formula Area of an ellipse fracx^2a^2 + fracy^2b^2 = 1$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is given by: textArea = pi a b$$\text{Area} = \pi a b$$ Area of a rhombus bounded by |x| + |y| = a$|x| + |y| = a$ is 2a^2$2a^2$. ### Core Logic The given curves form a bounded geometric area. Ellipse: x^2 + 4y^2 = 4 Rightarrow fracx^24 + fracy^21 = 1$x^2 + 4y^2 = 4 \Rightarrow \frac{x^2}{4} + \frac{y^2}{1} = 1$. Here, a = 2$a = 2$, b = 1$b = 1$. The region to be excluded is bounded by y = |x| - 1$y = |x| - 1$ and y = 1 - |x|$y = 1 - |x|$, which rearranges to |x| + |y| = 1$|x| + |y| = 1$. This forms a square/rhombus centered at the origin with vertices at (1, 0), (0, 1), (-1, 0), (0, -1)$(1, 0), (0, 1), (-1, 0), (0, -1)$. ### Step 1: Calculate Total and Excluded Areas Total Area of the Ellipse: textArea = pi (2)(1) = 2pi$$\text{Area} = \pi (2)(1) = 2\pi$$ Excluded Area (Rhombus |x| + |y| = 1$|x| + |y| = 1$): The rhombus consists of 4 identical right-angled triangles in each quadrant. Area of one triangle = frac12 times textbase times textheight = frac12 times 1 times 1 = frac12$\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times 1 = \frac{1}{2}$. Total excluded area = 4 times frac12 = 2$4 \times \frac{1}{2} = 2$. ### Step 2: Calculate Required Area

Required Area = Area of ellipse -$-$ Shaded Area = 2pi - 2$= 2\pi - 2$ = 2(pi - 1)$= 2(\pi - 1)$ ### Pattern Recognition Transform absolute value equations y = pm(|x| - a)$y = \pm(|x| - a)$ into |x| + |y| = a$|x| + |y| = a$ to instantly recognize a standard rhombus, allowing direct geometry formulas instead of integration. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Maths: Area Under Curves Class 11 Maths: Conic Sections