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 :
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
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