Question

If $x = a\left( {t\cos t - \sin t} \right),y = a\left( {t\sin t + \cos t} \right)$. Then $\dfrac{{dy}}{{dx}}$ is equal to
1. $ - \tan t$
2. $\tan t$
3. $\cot t$
4. $ - \cot t$

Answer 1

Hint: In this question, we are given two equations $x = a\left( {t\cos t - \sin t} \right),y = a\left( {t\sin t + \cos t} \right)$ and we have to find the value of $\dfrac{{dy}}{{dx}}$. First step is to differentiate both the equations with respect to $t$. Now, divide the derivative of the second value by the first one. You’ll get the solution.

Formula used:
Product rule of differentiation – According to the Product Rule, the derivative of a product of two functions is the first function multiplied by the derivative of the second function plus the derivative of the second function multiplied by the derivative of the first function. When taking the derivative of the quotient of two functions, the Product Rule must be used.
$\dfrac{d}{{dx}}\left( {pq} \right) = p\dfrac{{dq}}{{dx}} + q\dfrac{{dp}}{{dx}}$

Complete step by step solution: 
Given that,
$x = a\left( {t\cos t - \sin t} \right) - - - - - \left( 1 \right)$
$y = a\left( {t\sin t + \cos t} \right) - - - - - \left( 2 \right)$
Differentiate equation (1) with respect to $t$,
$\dfrac{{dx}}{{dt}} = a\left( {t\left( { - \sin t} \right) + \cos t - \cos t} \right)$
$\dfrac{{dx}}{{dt}} = - at\sin t - - - - - \left( 3 \right)$
Differentiate equation (2) with respect to $t$,
$\dfrac{{dy}}{{dt}} = a\left( {t\cos t + \sin t - \sin t} \right)$
$\dfrac{{dy}}{{dt}} = at\cos t - - - - - \left( 4 \right)$
Now, dividing equation (4) by (3)
It gives $\dfrac{{dy}}{{dx}} = - \cot t$
Hence, option (4) is the correct answer i.e., $ - \cot t$.

Note: The key concept involved in solving this problem is the good knowledge of Differentiation. Students must remember that differentiation is a technique for determining a function's derivative. Differentiation is a mathematical process that determines the instantaneous rate of change of a function based on one of its variables.