Question

In the curve $y=c{e}^{x/a},$ where $a,c\in R-\left\{0\right\},$ the

• A. subtangent is constant
• B. subnormal varies as the square of the ordinate
• C. tangent at $(x_1,y_1)$ on the curve intersects the $x-$axis at a distance of $(x_1-a)$ from the origin
• D. equation of the normal at the point where the curve cuts the $y-$axis is $cy+ax=c^2$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A subtangent is constant
B subnormal varies as the square of the ordinate
C tangent at $(x_1,y_1)$ on the curve intersects the $x-$axis at a distance of $(x_1-a)$ from the origin
D equation of the normal at the point where the curve cuts the $y-$axis is $cy+ax=c^2$

We have $y=e^{\frac{x}{a}}$
$\therefore \frac{dy}{dx}=\frac{c}{a}{e}^{\frac{x}{a}}=\frac{1}{a}y\Rightarrow \frac{y}{dy/dx}=a=\text{constant}$
$\Rightarrow$ Length of subtangent $=$ constant

Length of the subnormal $=y\frac{dy}{dx}=y\frac{y}{a}=\frac{y^2}{a}\propto \text{square of the ordinate}$

Equation of the tangent at $(x_1,y_1)$ is $y-y_1=\frac{y_1}{a}(x-x_1)$

This line meets the $x$-axis at a point given by $-y_1=\frac{y_1}{a}(x-x_1)$
$\Rightarrow x=x_1-a$

The curve meets the $y-$axis at $(0,c).$
Therefore, ${\left(\frac{dy}{dx}\right)}_{(0,c)}=\frac{c}{a}$

So, the equation of the normal at $(0,c)$ is
$y-c=-\frac{1}{c/a}(x-0)\Rightarrow ax+cy=c^2$