A particle moves along the curve `y = 4x^(2)+ 2` , then the point on the curve at which -y coordinates is changing 8 times as fast as the x-coordinate is

Options

A. `(2,18)`
B. ` (-1,6)`
C. ` (1,6)`
D. ` (0,2)`

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand the given curve
The particle moves along the curve defined by the equation:
\[ y = 4x^2 + 2 \]

### Step 2: Differentiate the equation with respect to time \( t \)
To find how \( y \) changes with respect to \( t \), we differentiate both sides of the equation with respect to \( t \):
\[ \frac{dy}{dt} = \frac{d}{dt}(4x^2 + 2) \]
Using the chain rule, we get:
\[ \frac{dy}{dt} = 8x \frac{dx}{dt} \]
This is our **Equation (1)**.

### Step 3: Set up the relationship given in the problem
According to the problem, the \( y \)-coordinate is changing 8 times as fast as the \( x \)-coordinate. This means:
\[ \frac{dy}{dt} = 8 \frac{dx}{dt} \]
This is our **Equation (2)**.

### Step 4: Equate the two expressions for \(\frac{dy}{dt}\)
From **Equation (1)** and **Equation (2)**, we can set them equal to each other:
\[ 8x \frac{dx}{dt} = 8 \frac{dx}{dt} \]

### Step 5: Simplify the equation
Assuming \(\frac{dx}{dt} \neq 0\), we can divide both sides by \(\frac{dx}{dt}\):
\[ 8x = 8 \]
Dividing both sides by 8 gives:
\[ x = 1 \]

### Step 6: Substitute \( x \) back into the curve equation to find \( y \)
Now, we substitute \( x = 1 \) back into the original curve equation to find the corresponding \( y \)-coordinate:
\[ y = 4(1)^2 + 2 = 4 \cdot 1 + 2 = 4 + 2 = 6 \]

### Step 7: State the required point
Thus, the required point on the curve is:
\[ (1, 6) \]

### Final Answer
The required point is \( (1, 6) \).

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