The area of the region bounded by the parabola $(y-2)^2=x-1$, the tangent to the parabola at the point $(2…
The area of the region bounded by the parabola $(y-2)^2=x-1$, the tangent to the parabola at the point $(2,3)$ and the $x$-axis is
3
6
9
12
Solution
Equation of tangent at $(2,3)$ to
$
\begin{aligned}
& (y-2)^2=x-1 \text { is } S_1=0 \\
& \Rightarrow x-2 y+4=0
\end{aligned}
$
Required Area $=$ Area of $\triangle O C B+$ Area of OAPD - Area of $\triangle P C D$
$
\begin{aligned}
& =\frac{1}{2}(4 \times 2)+\int_0^3\left(y^2-4 y+5\right) d y-\frac{1}{2}(1 \times 2) \\
& =4+\left[\frac{y^3}{3}-2 y^2+5 y\right]_0^3-1=4-9-18+15-1 \\
& =28-19=9 \text { sq. units }
\end{aligned}
$
$
\text { (or) }
$
$
\text { Area }=\int_0^3\left(2 y-4-y^2+4 y-5\right) d y=\int_0^3\left(-y^2+6 y-5\right) d y=-\int_0^3(3-y)^2 d y=\left[\frac{(y-3)^3}{3}\right]_0^3=\frac{27}{3}=9 \text { sq.units }
$
