 Hello and welcome to the session. Let us understand the following question today. If x and y are connected parametrically by the equations given below, without eliminating the parameter find d y by dx, we have x is equal to a cos theta plus theta sin theta and y is equal to a sin theta minus theta cos theta. Now let us write the solution. Given to us is x is equal to a multiplied by cos theta plus theta sin theta and y is equal to a multiplied by sin theta minus theta cos theta. Now consider x is equal to a multiplied by cos theta plus theta sin theta. Therefore dx by d theta is equal to a multiplied by minus sin theta plus theta cos theta plus sin theta. Now here minus sin theta gets answered with plus sin theta. So dx by d theta is equal to a theta cos theta. Now consider y is equal to a multiplied by sin theta minus theta cos theta. Therefore dy by d theta is equal to a multiplied by cos theta minus theta minus sin theta minus cos theta. Now this and this gets cancelled so it is equal to a theta sin theta. Now by chain rule dy by dx is equal to dy by d theta multiplied by d theta by dx which is equal to dy by d theta is equal to a theta sin theta multiplied by d theta by dx is equal to inverse of dx by d theta which is equal to 1 by a theta cos theta. Now we see that a gets cancelled with a theta gets cancelled with theta. So we are left with sin theta by cos theta which is equal to tan theta which is our dy by dx. Therefore the required answer is tan theta. I hope you understood the question bye and have a nice day.