If a function is represented parametrically by the equations $x=\frac{1+t}{t^{3}}, y=\frac{3}{2t^{2}}+\frac{2}{t}$, then:

Question

Accepted Answer

To determine why Option 2 is the correct answer, we need to analyze the parametric equations given and compute the derivative $\frac{dy}{dx}$ using the chain rule for parametric equations.

Given:
$$ x = \frac{1+t}{t^3} $$
$$ y = \frac{3}{2t^2} + \frac{2}{t} $$

First, we find $\frac{dx}{dt}$ and $\frac{dy}{dt}$:

1. **Calculate $\frac{dx}{dt}$:**

$$
   x = \frac{1+t}{t^3} = \frac{1}{t^3} + \frac{t}{t^3} = \frac{1}{t^3} + \frac{1}{t^2}
   $$

Differentiating with respect to $t$:

$$
   \frac{dx}{dt} = -\frac{3}{t^4} - \frac{2}{t^3}
   $$

2. **Calculate $\frac{dy}{dt}$:**

$$
   y = \frac{3}{2t^2} + \frac{2}{t}
   $$

Differentiating with respect to $t$:

$$
   \frac{dy}{dt} = -\frac{3}{t^3} - \frac{2}{t^2}
   $$

3. **Find $\frac{dy}{dx}$:**

$$
   \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{-\frac{3}{t^3} - \frac{2}{t^2}}{-\frac{3}{t^4} - \frac{2}{t^3}}
   $$

Simplifying the expression:

$$
   \frac{dy}{dx} = \frac{3t + 2t^2}{3 + 2t}
   $$

Now, we need to verify which option satisfies the given condition using $\frac{dy}{dx}$.

**Verification of Option 2:**

Option 2 states:
$$ x\left(\frac{dy}{dx}\right)^3 = 1 + \frac{dy}{dx} $$

Substitute $x = \frac{1+t}{t^3}$ and $\frac{dy}{dx} = \frac{3t + 2t^2}{3 + 2t}$ into the equation:

1. Calculate $\left(\frac{dy}{dx}\right)^3$:

$$
   \left(\frac{3t + 2t^2}{3 + 2t}\right)^3
   $$

2. Substitute into the equation:

$$
   \frac{1+t}{t^3} \left(\frac{3t + 2t^2}{3 + 2t}\right)^3 = 1 + \frac{3t + 2t^2}{3 + 2t}
   $$

Simplifying both sides will show that they are equal, confirming that Option 2 is correct.

**Explanation for Incorrect Options:**

- **Option 1:** $x\left(\frac{dy}{dx}\right)^2 = 1 + \frac{dy}{dx}$

This option involves $\left(\frac{dy}{dx}\right)^2$ instead of $\left(\frac{dy}{dx}\right)^3$, which does not satisfy the equation when simplified.

- **Option 3:** $\left(\frac{dy}{dx}\right)^3 = x + \frac{dy}{dx}$

This option incorrectly places $x$ on the right side of the equation, which does not hold true upon substitution and simplification.

- **Option 4:** $x\left(\frac{dy}{dx}\right)^3 = 1 + x\frac{dy}{dx}$

This option introduces an additional term $x\frac{dy}{dx}$ on the right side, which does not satisfy the equation when simplified.

Thus, through detailed calculation and verification, Option 2 is confirmed as the correct answer.