Question

In the curve $y={\mathrm{ce}}^{\frac{x}{a}}$, the

• A. The length of a sub-tangent is constant
• B. The length of a 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 $\mathrm{cy}+\mathrm{ax}=c^2$

Answer 1

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

equation of the normal at the point where the curve cuts y-axis is $\mathrm{cy}+\mathrm{ax}=c^2$

We have, $y=c{e}^{x/a}$
$\therefore \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{c}{a}{e}^{\frac{x}{a}}=\frac{1}{a}y\mathrm{or},\left(\frac{y}{\frac{\mathrm{dy}}{\mathrm{dx}}}\right|=\left(a\right|=\mathrm{constant}\mathrm{or},\mathrm{The}\mathrm{length}\mathrm{of}a\mathrm{Sub}-\mathrm{tangent}=\mathrm{constant}\mathrm{or},\mathrm{Length}\mathrm{of}a\mathrm{sub}-\mathrm{normal}=\left(y\frac{\mathrm{dy}}{\mathrm{dx}}\right|=\left(y\frac{y}{a}\right|=\frac{y^2}{\left(}\propto \left(\mathrm{square}\mathrm{of}\mathrm{the}\mathrm{ordinate}\right)\mathrm{Equation}\mathrm{of}\mathrm{the}\mathrm{tangent}\mathrm{at}\left(x_1,y_1\right)\mathrm{is}y-y_1=\frac{y_1}{a}\left(x-x_1\right)\mathrm{This}\mathrm{meets}\mathrm{the}x-\mathrm{axis}\mathrm{at}a\mathrm{point}\mathrm{given}\mathrm{by}-y_1=\frac{y_1}{a}\left(x-x_1\right)\mathrm{or}x-x_1=-a\Rightarrow x=x_1-a\mathrm{The}\mathrm{curve}\mathrm{meets}\mathrm{the}y-\mathrm{axis}\mathrm{at}\left(0,c\right).\therefore {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}_{\left(0,c\right)}=\frac{c}{a}\mathrm{So},\mathrm{the}\mathrm{equation}\mathrm{of}\mathrm{the}\mathrm{normal}\mathrm{at}(0,c)\mathrm{is}y-c=-\frac{1}{\frac{c}{a}}\left(x-0\right)\mathrm{or},\mathrm{ax}+\mathrm{cy}=c^2.$