Question

The area (in sq. units) bounded by the parabola $y=x^2-1$, the tangent at the point $(2,3)$ to it and the $y$-axis is:

• A. $\frac{8}{3}$
• B. $\frac{14}{3}$
• C. $\frac{56}{3}$
• D. $\frac{32}{3}$

Answer 1

The correct option is A $\frac{8}{3}$


Equation of the tangent at $(2,3)$ is $y-3=4(x-2)$
$[\because {\left(\frac{dy}{dx}\right)}_{(2,3)}=4]$
$\Rightarrow y=4x-5$
At $x=0,y=-5$
$\therefore$ Required area $=\text{area}\left(△ABC\right)-\text{area}\left(OABP\right)$
$=\frac{1}{2}\times 8\times 2-\underset{-1}{\overset{3}{\int }}\sqrt{y+1}dy$
$=8-\frac{2}{3}{\left[\right(y+1{)}^{\frac{3}{2}}]}_{-1}^3=8-\frac{16}{3}$
$=\frac{8}{3}$ sq. units