QJEEM24Ma82d2: JEE Main 2024 Mathematics - Class 12 JEEM24Ma82d2 Solution: The area of the region enclosed by the parabola , the line and…

Solving the equations

$\begin{array}{r}(y-2)^2=x-1 \text { and } x-2 y+4=0 \\x=2(y-2)\end{array}$

$\begin{aligned}& \frac{x^2}{4}=x-1 \\& x^2-4 x+4=0 \\& (x-2)^2=0 \\& x=2\end{aligned}$

Exclose area (w.r.t. y-axis) $$=\underset{0}{\overset{3}{\int }}xdy-\text{ Area of }\mathrm{\Delta }$$.

$$\begin{array}{rl}& =\underset{0}{\overset{3}{\int }}((y-2{)}^2+1)dy-\frac{1}{2}\times 1\times 2\\ & =\underset{0}{\overset{3}{\int }}(y^2-4y+5)dy-1\\ & ={[\frac{y^3}{3}-2y^2+5y]}_0^3-1\\ & =9-18+15-1=5\end{array}$$