For the curve $y=b{e}^{x/a}$, the subtangent is of constant length and the subnormal varies as

Sol. The equation of given curve is $y=b{e}^{x/a}$

Let us consider a point $\left(x_1,y_1\right)$ on the curve.

Then, we have $y_1=b{e}^{x_1/a}$

Differentiating the curve $y=b{e}^{x/a}$ w.r.t. $x$, we get

$\frac{dy}{dx}=b{e}^{x/a}·\frac{1}{a}$

$\therefore {\left(\frac{dy}{dx}\right)}_{\left(x_1,y_1\right)}=\frac{b}{a}{e}^{x_1/a}$

Thus, the length of subtangent $=\left|y_1{\left(\frac{dx}{dy}\right)}_{\left(x_1,y_1\right)}\right|$

$=\left|y_1·\frac{a}{b{e}^{x_1/a}}\right|=\left|b{e}^{x_1/a}·\frac{a}{b{e}^{x_1/a}}\right|$

$=a$ (constant)

[using Eq. (i) and (ii)]

$\Rightarrow$ Subtangent is of constant length $a$.

Again, length of subnormal $=\left|y_1{\left(\frac{dy}{dx}\right)}_{\left(x_1,y_1\right)}\right|$

$=\left|b{e}^{x_1/a}·\frac{b{e}^{x_1/a}}{a}\right|=\frac{1}{a}{\left(b{e}^{x_1/a}\right)}^2=\frac{1}{a}{y}^2$ 

[using Eq. (i)] 

Therefore, subnormal varies as the square of ordinate.

Hence, the correct option is $1.$