JEE Mathematics 2026 Question with Solution

Given: The ellipse is $x^2+4y^2=4$ and the answer must correspond to the area inside the ellipse and outside the excluded region.

Find: The required area.

From the solution working, the wording about the region bounded by $y=x-1$ and $y=1-x$ is treated as ambiguous. The solution interprets the excluded region as the rhombus bounded by the four lines $y=x-1$, $y=1-x$, $y=-x-1$, and $y=x+1$, whose equation is $|x|+|y|=1$.

First, write the ellipse in standard form:

So, its semi-axes are $a=2$ and $b=1$. Therefore, the area of the ellipse is

Now consider the excluded rhombus with vertices $(1,0), (0,1), (-1,0), (0,-1)$. Its diagonals are

Hence its area is

Therefore, the required area is

Therefore, the correct option is C.

Given: the solution itself notes that the statement about the region bounded by the two lines is ambiguous.

Find: The interpretation used to match the answer.

The solution explicitly says the answer choices suggest a result of the form area of ellipse minus a simple geometric area. Since the ellipse has area $2\pi$ and the accepted answer is $2\pi-2$, the excluded area must be $2$.

A natural region of area $2$ formed by the relevant family of lines is the rhombus $|x|+|y|=1$. This rhombus has diagonals both equal to $2$, so its area is $2$. Subtracting this from the ellipse area gives $2(\pi-1)$.

The solution also notes a discrepancy in wording: only two lines are named in the question text, while the interpreted bounded region actually uses four lines. Nevertheless, the solution concludes that this is the intended reading and selects option C.