Question

In the curve $y=c{e}^{x/a}$, the

• A. sub-tangent is constant
• B. sub-normal 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 $y$-axis is $cy+ax=c^2$

Answer 1

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

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

The given curve is $y=c{e}^{\frac{x}{a}}$
Let point $P(x_1,y_1)$ be on the curve.
$\Rightarrow y_1=c{e}^{\frac{x_1}{a}}$ .....(i)
$\frac{dy}{dx}=c{e}^{\frac{x}{a}}.\frac{1}{a}$
Slope of tangent at $(x_1,y_1)$ is $m={\left(\frac{dy}{dx}\right)}_{(x_1,y_1)}=\frac{c}{a}{e}^{\frac{x_1}{a}}=\frac{y_1}{a}$
Thus, the length of subtangent$=\frac{y_1}{m}=a$ (constant)
$\Rightarrow$ Subtangent is of constant length $a$.
Again, length of subnormal$=|y_{1}m|$
$=\frac{1}{a}{\left(c{e}^{\frac{x_1}{a}}\right)}^2=\frac{1}{a}{y}^2$
Therefore, subnormal varies as the square of ordinate.
Equation of tangent at $(x_1,y_1)$ is
$y-y_1=\frac{y_1}{a}(x-x_1)$
Since, tangent intersects x-axis i.e. $y=0$
$\Rightarrow x=x_1-a$
Now, again since the curve intersect y-axis, i.e $x=0$
$\Rightarrow y=c$
So, the slope of normal at point (0,c) is $-\frac{a}{c}$
Equation of normal at (0,c) is
$y-c=-\frac{a}{c}(x-0)$
$cy+ax=c^2$