Find `dy/dx`, if x and y are connected parametrically by the equations, given below without eliminating the parameter:
`x=acostheta,y=bsintheta`

Answer

To find \(\frac{dy}{dx}\) given the parametric equations \(x = a \cos \theta\) and \(y = b \sin \theta\), we will follow these steps:

### Step 1: Differentiate \(x\) and \(y\) with respect to \(\theta\)
We start by differentiating both \(x\) and \(y\) with respect to the parameter \(\theta\).

1. Differentiate \(x = a \cos \theta\):
\[ \frac{dx}{d\theta} = -a \sin \theta \]

2. Differentiate \(y = b \sin \theta\):
\[ \frac{dy}{d\theta} = b \cos \theta \]

### Step 2: Use the chain rule to find \(\frac{dy}{dx}\)
Using the chain rule, we can express \(\frac{dy}{dx}\) in terms of \(\frac{dy}{d\theta}\) and \(\frac{dx}{d\theta}\):
\[ \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} \]

### Step 3: Substitute the derivatives into the formula
Now, we substitute the derivatives we found in Step 1 into the formula from Step 2:
\[ \frac{dy}{dx} = \frac{b \cos \theta}{-a \sin \theta} \]

### Step 4: Simplify the expression
We can simplify the expression:
\[ \frac{dy}{dx} = -\frac{b}{a} \cdot \frac{\cos \theta}{\sin \theta} \]
Since \(\frac{\cos \theta}{\sin \theta} = \cot \theta\), we can rewrite it as:
\[ \frac{dy}{dx} = -\frac{b}{a} \cot \theta \]

### Final Result
Thus, the final result for \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = -\frac{b}{a} \cot \theta \]