Question

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

Answer 1

The correct option is C $9$ sq. units

Given parabola is $(y-2{)}^2=x-1$

$\Rightarrow \frac{dy}{dx}=\frac{1}{2(y-2)}$

When $y=3,x=2$

$\therefore \frac{dy}{dx}=\frac{1}{2}$.

Tangent at $(2,3)$ is $y-3=\frac{1}{2}(x-2)\Rightarrow x-2y+4=0$

$\therefore$ required area

$={\int }_0^3\left(\right(y-2{)}^2+1)dy-{\int }_0^3(2y-4)dy$

$={|\frac{(y-2{)}^3}{3}+y|}_0^3-|y^2-4y{|}_0^3$

$=\frac{1}{3}+3+\frac{8}{3}-(9-12)=9$ sq. units.