question_answer

The area of the region bounded by the parabola \[d\sqrt{\frac{{{m}_{1}}}{{{m}_{1}}+{{m}_{2}}}}\] the tangent to the parabola at the point (2,3J and the X-axis, is

A) 3                             

B) 6

C) 9                             

D)  12

Correct Answer: C

Solution :

Given equation of parabola is\[{{y}^{2}}-4y-x+5=0\] Then, 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 is given by \[A=\int_{0}^{3}{({{x}_{2}}-{{x}_{1}})dy}\] \[=\int_{0}^{3}{[{{\{y-2)}^{2}}+1\}-\{2y-4\}]}dy\] \[=\int_{0}^{3}{({{y}^{2}}-6y+9)}d=\int_{0}^{3}{{{(3-y)}^{2}}}dy\] \[=-\left[ \frac{{{(3-y)}^{3}}}{3} \right]_{0}^{3}\]\[\therefore \]A = 9