A particle moves along the curve $6y=x^{3}+2$. The number of points on the curve at which the y-co-ordinate is changing eight times the x-co-ordinate is:

Question

Accepted Answer

To determine the number of points on the curve $6y = x^3 + 2$ where the y-coordinate is changing eight times the x-coordinate, we need to analyze the relationship between the rates of change of $x$ and $y$.

1. **Differentiating the Curve Equation:**

The given curve is $6y = x^3 + 2$. To find the rate of change of $y$ with respect to $x$, we differentiate both sides with respect to $x$:

$$
   \frac{d}{dx}(6y) = \frac{d}{dx}(x^3 + 2)
   $$

This gives:

$$
   6 \frac{dy}{dx} = 3x^2
   $$

Solving for $\frac{dy}{dx}$, we get:

$$
   \frac{dy}{dx} = \frac{3x^2}{6} = \frac{x^2}{2}
   $$

2. **Condition for the Rate of Change:**

We are given that the y-coordinate is changing eight times the x-coordinate. Mathematically, this condition can be expressed as:

$$
   \frac{dy}{dx} = 8
   $$

Substituting the expression for $\frac{dy}{dx}$ from the differentiation, we have:

$$
   \frac{x^2}{2} = 8
   $$

Solving for $x^2$, we multiply both sides by 2:

$$
   x^2 = 16
   $$

Taking the square root of both sides, we find:

$$
   x = \pm 4
   $$

3. **Finding Corresponding y-coordinates:**

For each value of $x$, we substitute back into the original curve equation to find the corresponding $y$-coordinates.

- For $x = 4$:

$$
     6y = 4^3 + 2 = 64 + 2 = 66 \quad \Rightarrow \quad y = \frac{66}{6} = 11
     $$

- For $x = -4$:

$$
     6y = (-4)^3 + 2 = -64 + 2 = -62 \quad \Rightarrow \quad y = \frac{-62}{6} = -\frac{31}{3}
     $$

4. **Counting the Points:**

We have found two distinct points on the curve where the condition is satisfied: $(4, 11)$ and $(-4, -\frac{31}{3})$.

Therefore, there are exactly 2 points on the curve where the y-coordinate changes eight times the x-coordinate, which supports Option 3 as the correct answer.

**Explanation of Incorrect Options:**

- **Option 1 (4 points):** This option is incorrect because we only found two points that satisfy the given condition, not four.

- **Option 2 (3 points):** This option is incorrect because there are only two solutions to the equation $x^2 = 16$, leading to two points, not three.

- **Option 4 (1 point):** This option is incorrect because we identified two distinct points that satisfy the condition, not just one.

Thus, the correct answer is indeed Option 3, as there are exactly 2 points on the curve that meet the specified condition.