The area of the region bounded
Question:

The area of the region bounded by the parabola $(y-2)^{2}=(x-1)$, the tangent to it at the point whose ordinate is 3 and the $x$-axis is :

1. 9

2. 10

3. 4

4. 6

Correct Option: 1

Solution:

$y=3 \Rightarrow x=2$

Point is $(2,3)$

Diff. w.r.t $\quad x$

$2(y-2) y^{\prime}=1$

$\Rightarrow y^{\prime}=\frac{1}{2(y-2)}$

$\Rightarrow \mathrm{y}_{(2,3)}^{\prime}=\frac{1}{2}$

$\Rightarrow \frac{y-3}{x-2}=\frac{1}{2} \Rightarrow x-2 y+4=0$

Area $=\int_{0}^{3}\left((y-2)^{2}+1-(2 y-4)\right) d y$

$=9 \mathrm{sq} .$ units