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Area Under the Curve

Area Inside Ellipse, Outside Rhombus

nta_pyq_2026_jan

Grade 12

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:

$2\pi-1$

$3(\pi-1)$

$2\pi-\dfrac{1}{2}$

$2(\pi-1)$

Step-by-Step Solution

Key Concept: Ellipse: $\frac{x^2}{4}+y^2=1$ with $a=2,b=1$, area $=\pi ab=2\pi$. Curves $y=|x|-1$ and $y=1-|x|$ intersect at $(\pm1,0)$ and $(0,\pm1)$, forming a rhombus with vertices $(0,1),(1,0),(0,-1),(-1,0)$.

Ellipse area $=2\pi$, rhombus area $=2$. Required $=2(\pi-1)$.

Correct Answer: 4