JEE 2023 — Mathematics PYQ

Step 1: Express the curves in terms of $y$

• Parabola: $y^2+4x=4⟹4x=4-y^2⟹x=1-\frac{y^2}{4}$

• Line: $y-2x=2⟹2x=y-2⟹x=\frac{y-2}{2}$

Step 2: Find the Points of Intersection

To find where the curves meet, set the $x$ values equal to each other:

Multiply the entire equation by $4$ to clear the fractions:

Rearrange into a quadratic equation:

Factor the quadratic:

The points of intersection are at $y=-4$ and $y=2$.

Step 3: Calculate the Area

We integrate with respect to $y$ from $-4$ to $2$:

Simplify the integrand:

Perform the integration:

Apply the limits:

• Upper limit ($y=2$): $2(2)-\frac{4}{4}-\frac{8}{12}=4-1-\frac{2}{3}=\frac{7}{3}$

• Lower limit ($y=-4$): $2(-4)-\frac{16}{4}-\frac{-64}{12}=-8-4+\frac{16}{3}=-12+\frac{16}{3}=-\frac{20}{3}$

Correct Option: (A) 9