question_answer

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                 \[A=\int_{0}^{3}{({{x}_{2}}-{{x}_{1}})}dx\] \[\Rightarrow \]               \[A=\int_{0}^{3}{[\{{{(y-2)}^{2}}+1\}-\{2y-4\}]dy}\] \[\Rightarrow \]               \[A=\int_{0}^{3}{({{y}^{2}}-6y+9)dy}\]                 \[=\int_{0}^{3}{{{(3-y)}^{2}}}dy=-\left[ \frac{{{(3-y)}^{3}}}{3} \right]_{0}^{3}=9\]