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
A)\[3\]
B)\[6\]
C)\[9\]
D)\[12\]
Correct Answer:
C
Solution :
Given, equation of the parabola is \[{{(y-2)}^{2}}=(x-1)\] or \[{{y}^{2}}-4y-x+5=0\] The equation of tangent at \[(2,\,\,3)\] is \[3y-2(y+3)-\frac{(x+2)}{2}+5=0\] \[\Rightarrow \] \[2y-x-4=0\] \[\therefore \]Required area \[A\] is given by \[A=\int_{0}^{3}{({{x}_{2}}-{{x}_{1}})}dx\] \[\Rightarrow \] \[A=\int_{0}^{3}{[\{{{(y-2)}^{2}}+1\}-\{2y-4\}]dy}\] \[\Rightarrow \] \[A=\int_{0}^{3}{({{y}^{2}}-6y+9)dy}\] \[=\int_{0}^{3}{{{(3-y)}^{2}}}dy=-\left[ \frac{{{(3-y)}^{3}}}{3} \right]_{0}^{3}=9\]