Consider the parabola `y = x^(2) + 7x + 2` and the straight line y = 3x -3.
What are the coordinates of the point on the parabola which is closest to the straight line?

Options

A. (0, 2)
B. (-2, -8)
C. (-7, 2)
D. (1, 10)

Answer

To find the coordinates of the point on the parabola \( y = x^2 + 7x + 2 \) that is closest to the straight line \( y = 3x - 3 \), we can follow these steps:

### Step 1: Identify the equations
The equations given are:
- Parabola: \( y = x^2 + 7x + 2 \)
- Line: \( y = 3x - 3 \)

### Step 2: Find the slope of the tangent to the parabola
To find the slope of the tangent to the parabola at any point, we differentiate the equation of the parabola:
\[ \frac{dy}{dx} = 2x + 7 \]

### Step 3: Set the slope of the tangent equal to the slope of the line
The slope of the line \( y = 3x - 3 \) is 3. For the point on the parabola to be closest to the line, the slope of the tangent to the parabola must equal the slope of the line:
\[ 2x + 7 = 3 \]

### Step 4: Solve for \( x \)
Now, we solve the equation:
\[ 2x + 7 = 3 \\ 2x = 3 - 7 \\ 2x = -4 \\ x = -2 \]

### Step 5: Substitute \( x \) back into the parabola to find \( y \)
Now that we have \( x = -2 \), we can find the corresponding \( y \) value by substituting \( x \) back into the equation of the parabola:
\[ y = (-2)^2 + 7(-2) + 2 \\ y = 4 - 14 + 2 \\ y = -8 \]

### Step 6: Write the coordinates of the point
The coordinates of the point on the parabola that is closest to the line are:
\[ (-2, -8) \]

### Final Answer
The coordinates of the point on the parabola which is closest to the straight line are:
\[ \boxed{(-2, -8)} \]