Question

The equation to the tangent to the curve $y={be}^{-x/a}$ at the point where it crosses the $y$-axis is

• A. $ax+by=1$
• B. $\frac{x}{a}-\frac{y}{b}=1$
• C. $\frac{x}{a}+\frac{y}{b}=1$
• D. $ax-by=1$

Answer 1

The correct option is C $\frac{x}{a}+\frac{y}{b}=1$

Slope of the tangent to $f$ is $f^{\prime }$

So, $y^{\prime }=\frac{d}{dx}\left(b{e}^{-x/a}\right)=b\left(\frac{-1}{a}\right)e^{-x/a}$

When it crosses $y-axis$, point is $(0,b)$.

At $(0,b)$ slope is $-\frac{b}{a}$ and is passes through point $(0,b)$

Then, the equation of line is $y-b=-\frac{b}{a}(x-0)$

$⟹ay-ab=-bx$

$⟹\frac{x}{a}+\frac{y}{b}=1$ is the equation of tangent.