Question

Length of the subtangent at $\left( {{x}_{l}},{{y}_{l}} \right)$ on ${{x}^{n}}{{y}^{m}}={{a}^{m+n}}$ , $m,n>0$ , is.
(a) $\dfrac{n}{m}{{x}_{l}}$
(b) $\dfrac{m}{n}\left| {{x}_{l}} \right|$
(c) $\dfrac{n}{m}\left| {{y}_{l}} \right|$
(d) $\dfrac{n}{m}\left| {{x}_{l}} \right|$

Answer 1

Hint:For solving this question first we will simplify the given equation by taking log on both sides then we will differentiate it with respect to $x$ and calculate the value of $\dfrac{dy}{dx}$ . Then, we will directly find the length of the subtangent from its formula.

Complete step-by-step answer:
Given: We have to find the length of subtangent at $\left( {{x}_{l}},{{y}_{l}} \right)$ on ${{x}^{n}}{{y}^{m}}={{a}^{m+n}}$ , $m,n>0$ .
Now, we know that length of subtangent for any curve $y=f\left( x \right)$ at a point $\left( {{x}_{1}},{{y}_{1}} \right)$ on the curve is equal to ${{\left| y\dfrac{dx}{dy} \right|}_{\left( {{x}_{1}},{{y}_{1}} \right)}}$ . First, we will solve for $\dfrac{dy}{dx}$ then we will find the length of subtangent using ${{\left| y\dfrac{dx}{dy} \right|}_{\left( {{x}_{1}},{{y}_{1}} \right)}}$ .
The equation of the curve is ${{x}^{n}}{{y}^{m}}={{a}^{m+n}}$ . Now, take log to the base $e$ on both sides. Then,
$\begin{aligned} & {{x}^{n}}{{y}^{m}}={{a}^{m+n}} \\\ & \Rightarrow \log \left( {{x}^{n}}{{y}^{m}} \right)=\log \left( {{a}^{m+n}} \right) \\\ & \Rightarrow \log \left( {{x}^{n}} \right)+\log \left( {{y}^{m}} \right)=\left( m+n \right)\log a \\\ & \Rightarrow n\log x+m\log y=\left( m+n \right)\log a \\\ & \Rightarrow n\log x+m\log y=( m+n )\log a \\\ \end{aligned}$
Now, in the above equation $a$ is a constant so, differentiating the above equation with respect to $x$ . Then,
$\Rightarrow \dfrac{d\left( n\log x \right)}{dx}+\dfrac{d\left( m\log y \right)}{dx}=\dfrac{d\left( \left( m+n \right)\log a \right)}{dx} \\\$
We know that $(\dfrac{d\left( log x \right)}{dx}={\dfrac{1}{x}})$
Then we can write,
$\Rightarrow \dfrac{n}{x}+\dfrac{m}{y}\dfrac{dy}{dx}=0 \\\ \Rightarrow \dfrac{m}{y}\dfrac{dy}{dx}=-\dfrac{n}{x} \\\ \Rightarrow \dfrac{y}{m}\dfrac{dx}{dy}=-\dfrac{x}{n} \\\ \Rightarrow y\dfrac{dx}{dy}=-\dfrac{m}{n}x \\\ \Rightarrow \left| y\dfrac{dx}{dy} \right|=\left| -\dfrac{m}{n}x \right| \\\ \Rightarrow \left| y\dfrac{dx}{dy} \right|=\dfrac{m}{n}\left| x \right| \\\$
Now, from the above calculation, we can say that value of ${{\left| y\dfrac{dx}{dy} \right|}_{\left( {{x}_{1}},{{y}_{1}} \right)}}$ for the given curve will be value of $\dfrac{m}{n}\left| x \right|$ at $\left( {{x}_{l}},{{y}_{l}} \right)$ . Then, length of subtangent $=\dfrac{m}{n}\left| {{x}_{l}} \right|$ .

Hence, option (b) is the correct option.

Note: Here if the student directly differentiates the given equation then more calculation will be there. And the student should not confuse the formula of subtangent with the formula of subnormal. Moreover, the student should proceed stepwise and avoid calculation mistakes while solving to get the correct answer.