If x= 2at , y= 2a , where t is parameter, then `(dy)/(dx) = 1/t`. (True/False)

Answer

To determine whether the statement \((dy)/(dx) = 1/t\) is true or false given \(x = 2at\) and \(y = 2a\), we will follow these steps:

### Step 1: Write down the equations
We have:
\[ x = 2at \]
\[ y = 2a \]

### Step 2: Differentiate both equations with respect to \(t\)
Differentiate \(x\) with respect to \(t\):
\[ \frac{dx}{dt} = \frac{d}{dt}(2at) = 2a \]

Now differentiate \(y\) with respect to \(t\):
\[ \frac{dy}{dt} = \frac{d}{dt}(2a) = 0 \]
(since \(2a\) is a constant).

### Step 3: Use the chain rule to find \(\frac{dy}{dx}\)
Using the chain rule, we have:
\[ \frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dt}{dx} \]

### Step 4: Substitute the values we found
We already found:
\[ \frac{dy}{dt} = 0 \]
And we need to find \(\frac{dt}{dx}\). We can find \(\frac{dt}{dx}\) by taking the reciprocal of \(\frac{dx}{dt}\):
\[ \frac{dt}{dx} = \frac{1}{\frac{dx}{dt}} = \frac{1}{2a} \]

Now substituting into the equation for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = 0 \cdot \frac{1}{2a} = 0 \]

### Step 5: Compare with the given statement
The statement claims that \(\frac{dy}{dx} = \frac{1}{t}\). We found that:
\[ \frac{dy}{dx} = 0 \]

Since \(0 \neq \frac{1}{t}\) (for any \(t \neq 0\)), the statement is **False**.

### Final Answer:
The statement \((dy)/(dx) = 1/t\) is **False**.

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