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 x = 3, and the x-axis is

A) 7 sq. units         

B) 6 sq. units

C) 9 sq. units         

D) None of these

Correct Answer: C

Solution :

[c] 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 (2, 3) is \[y-3=\frac{1}{2}(x-2)\] Or \[x-2y+4=0\] \[\therefore \] Required area \[=\int_{0}^{3}{\left( {{(y-2)}^{2}}+1 \right)}dy-\int_{0}^{3}{(2y-4)dy}\] \[=\left| \frac{{{(y-2)}^{3}}}{3}+y \right|_{0}^{3}-|{{y}^{2}}-4y|_{0}^{3}\] \[=\frac{1}{3}+3+\frac{8}{3}-(9-12)=9\] sq. units.