Question

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. 6 sq units
• B. 9 sq units
• C. 12 sq units
• D. 3 sq units

Answer 1

The correct option is B 9 sq units

Given,

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

$\Rightarrow y^{\prime }=\frac{1}{2(y-2)}$

The equation of tangent to the parabola at $(2,3)$ is,

$y-3=\frac{1}{2}(x-2)$

$\Rightarrow x-2y+4=0$

The area of bonded region is:

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

$={\int }_0^3(y^2-6y+9)dy$

$={\int }_0^3(y-3{)}^{2}dy$

$={\int }_0^3{t}^{2}dt$

$={\left[\frac{t^3}{3}\right]}_0^3$

$=\frac{3^3}{3}=9$