Question

Let the equation of the curve be $x = a\left( {t + \sin t} \right),y = a(1 - \cos t)$ . Choose the correct option for the length of the subtangent at $t$ from the options given below.

1. $2a\sin \left( {\dfrac{t}{2}} \right)$
2. $2a{\sin ^3}\left( {\dfrac{t}{2}} \right)\sec \left( {\dfrac{t}{2}} \right)$
3. $a\sin t$
4. $2a\sin \left( {\dfrac{t}{2}} \right)\tan \left( {\dfrac{t}{2}} \right)$

Answer 1

Hint : Given curve is not a direct curve. It has a parameter $t$ . We have the formula for the length of the subtangent for the given curve at the given point. We just need to substitute the values required for the formula and then simplify the result to get the required answer.

Complete step-by-step answer :
Given that equation of the curve as,
$x = a\left( {t + \sin t} \right),y = a(1 - \cos t)$,
Now we need to find the length of the subtangent.
At first, we should have the point of tangency.
Given that the point of tangency is at $t$
So, the point of tangency is $\left( {{x_1},{y_1}} \right) = \left( {a\left( {t + \sin t} \right),a\left( {1 - \cos t} \right)} \right)$
We have the formula for the length of subtangent that is $\left| {\dfrac{{{y_1}}}{m}} \right|$
Where $m$ is the slope of the tangent
That is, $m = \dfrac{{dy}}{{dx}}$
So, length of subtangent $= \left| {{y_1}\dfrac{{dx}}{{dy}}} \right|$ .
Since the parameter is t we can write the formula as $= \left| {{y_1}\dfrac{{\dfrac{{dx}}{{dt}}}}{{\dfrac{{dy}}{{dt}}}}} \right|$
We have, ${y_1} = a\left( {1 - \cos t} \right)$
Now we just need to find the values of $\dfrac{{dx}}{{dt}},\dfrac{{dy}}{{dt}}$ .
Consider $\dfrac{{dx}}{{dt}}$ ,
We get,
$\dfrac{{dx}}{{dt}} = a\left( {1 + \cos t} \right)$.
Now consider $\dfrac{{dy}}{{dt}}$,
We get,
$\dfrac{{dy}}{{dt}} = a\sin t$
Now let us substitute all the values we got in the formula.
Length of subtangent$= \left| {{y_1}\dfrac{{dx}}{{dy}}} \right| = \left| {a(1 - \cos t)\dfrac{{\left( {a\left( {1 + \cos t} \right)} \right)}}{{a\sin t}}} \right|$,
Let simplify this by substituting half-angle formulas
$\Rightarrow \left| {{y_1}\dfrac{{dx}}{{dy}}} \right| = \left| {a\left( {2{{\sin }^2}\left( {\dfrac{t}{2}} \right)} \right)\dfrac{{\left( {a\left( {2{{\cos }^2}\left( {\dfrac{t}{2}} \right)} \right)} \right)}}{{a\left( {2\sin \dfrac{t}{2}\cos \dfrac{t}{2}} \right)}}} \right|$
By canceling all the possible cancelations, we get
$\Rightarrow \left| {{y_1}\dfrac{{dx}}{{dy}}} \right| = \left| {a\left( {2\sin \left( {\dfrac{t}{2}} \right)\cos \dfrac{t}{2}} \right)} \right|$
Observe that it has a half-angle formula again let us substitute it.
We get,
$\Rightarrow \left| {{y_1}\dfrac{{dx}}{{dy}}} \right| = \left| {a\sin t} \right|$
Therefore the length of the required subtangent is $a\sin t$.
So, the correct option is 3.
So, the correct answer is “Option 3”.

Note : This is a very good problem. There are many ways of getting errors in this problem. We must be careful while solving the problem. Calculations are heavy in this solving process. It may ease a little bit if we solve and get the slope first and then substitute it in the formula. The formulae are the key to this problem because if we do not know the formulae it takes more time to solve.