Question

The area bounded by the curve $y=x^2-1$, tangent to it at $\left(2,3\right)$ and Y-axis.

• A. $\frac{2}{3}$
• B. $\frac{4}{3}$
• C. $\frac{8}{3}$
• D. $1$

Answer 1

The correct option is C

$\frac{8}{3}$

Explanation of correct answer :

Finding the area bounded by the curve:

Equation of tangent at $\left(2,3\right)$ is

$$\begin{gathered} \left(\frac{y+3}{2}\right)=2x-1 \\ \Rightarrow y+3=4x-2 \\ \Rightarrow 4x-y=5 \end{gathered}$$

Required area,

$$\begin{gathered} =area\left(PAB\right)-{\int }_{-1}^3{x}_{parabola}dy \\ =\frac{1}{2}\times 2\times 8-{\int }_{-1}^3\sqrt{y+1}dy \\ =8-\frac{2}{3}{\left({\left(y+1\right)}^{\frac{3}{2}}\right)}_{-1}^3 \\ =8-\frac{16}{3} \\ =\frac{8}{3} \end{gathered}$$

Thus, the area is $\frac{8}{3}$sq. units

Hence, the correct answer is option(C).