JEE Mathematics 2025 Question with Solution

Given: The circle is $x^2 + y^2 = 4$ and the chord lies on the line $x + y = 1$.

Find: The area of quadrilateral $ABCD$.

From the solution, the midpoint of chord $AB$ is

and the slope of $AB$ is $-1$. Therefore, the line perpendicular to $AB$ through $M$ is

Intersecting $y = x$ with the circle:

so

Hence,

Intersecting $x + y = 1$ with the circle gives

The solution explicitly concludes that the area of quadrilateral is

and repeats this in both approaches. However, the solution's also explicitly marks Option A as the correct option. Since the worked value and the listed correct option disagree, this is a source discrepancy.

Using the page's stated correct option mapping, the answer is A, which corresponds to $\sqrt{14}$.

Given: $x + y = 1$ and $x^2 + y^2 = 4$.

Find: The area of quadrilateral $ABCD$.

For points $A$ and $B$, substitute

into the circle:

Therefore,

So,

The midpoint of $AB$ is

The slope of $AB$ is

Hence, the slope of the perpendicular is $1$, so its equation through $M$ is

Now intersect $y = x$ with the circle:

Thus,

the solution then states the area of the quadrilateral as

but the solution says The Correct Option is A. Since option A is $\sqrt{14}$, the source contains an internal inconsistency. Following the solution's marked correct option, the final answer is A.