Question

Tangent is drawn from $(1,0)$ to $y=e^x$, then the area bounded between the coordinate axes and the tangent is equal to -

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

Answer 1

Answer: (C)

$\frac{e^2}{2}$

Let $P(t,e^t)$ be any point on $y=e^x$

Slope of tangent at any point P on $y=e^x$ is $y^{\prime }\left(t\right)=e^t$ ........ $\left(1\right)$

Since the tangent passes through $(1,0)$

Therefore, slope of tangent is given by two point form i.e. $\frac{y_2-y_1}{x_2-x_1}=\frac{e^t}{t-1}$

From $\left(1\right)$: $\frac{e^t}{t-1}=e^t$

$\Rightarrow t=2$

Therefore, equation of tangent is $y-0=e^2(x-1)$

$\Rightarrow x+\frac{y}{-e^2}=1$

It is in intercept form $\frac{x}{a}+\frac{y}{b}=1$

Therefore, area with co-ordinate axis is $A=\frac{|a||b|}{2}=\frac{e^2}{2}$