question_answer

The area of the figure bounded by the parabola\[{{(y-2)}^{2}}=x-1\], the tangent to it at the point with the ordinate 3 and the x-axis is

A)  3                                            

B)  6

C)  9                            

D)         None of these

Correct Answer: C

Solution :

Given parabola, is                 \[{{(y-2)}^{2}}=x-1\] \[\Rightarrow \]                        \[\frac{dy}{dx}=\frac{1}{2(y-2)}\] when,\[y=3,\,\,x=2\] \[\therefore \]  \[\frac{dy}{dx}=\frac{1}{2(3-2)}=\frac{1}{2}\] Tangent at\[(x,\,\,3)\]is                 \[y-3=\frac{1}{2}\,(x-2)\]                 \[\Rightarrow \]               \[x-2y+4=0\] \[\therefore \]Required area \[\int_{0}^{3}{\{{{(y-2)}^{2}}+1\}dy-\int_{0}^{3}{(2y-4)dy}}\] \[=\left[ \frac{{{(y-2)}^{3}}}{3}+y \right]_{0}^{3}-[{{y}^{2}}-4y]_{0}^{3}\] \[=\frac{1}{3}+3+\frac{8}{3}-(9-12)=9\]