Question

From $(0,0),$ a tangent is drawn to the curve $y=e^x.$ Then the area bounded by the tangent, $y=e^x$ and $y-$axis is

• A. $2e$
• B. $e$
• C. $\frac{e}{2}$
• D. $\frac{e}{2}-1$

Answer 1

The correct option is D $\frac{e}{2}-1$


Let $(a,e^a)$ be the point where tangent touches $y=e^x$.
Then slope at $(a,e^a)$ is $e^a$
Equation of tangent at $(a,e^a)$ is $(y-e^a)=e^a(x-a)$
But the tangent passes through $(0,0)$
$\Rightarrow a=1$
So, equation of tangent is $y=ex$

Required area
$=\underset{0}{\overset{1}{\int }}(e^x-ex)dx$
$={[e^x-\frac{e{x}^2}{2}]}_0^1=(e-\frac{e}{2})-e^0=\frac{e}{2}-1$