Question

Let the shortest distance between the line y=x-3 and the parabola y=x²+3x+2 is d, then d² is ________.

Answer 1

8

Answer 2

To find the shortest distance between the line y = x - 3 and the parabola y = x² + 3x + 2, we follow these steps:

1. Identify the slope of the given line:
   The equation of the line is y = x - 3.
   The slope of this line (m_L) is 1.

2. Find the derivative of the parabola equation:
   The equation of the parabola is y = x² + 3x + 2.
   Differentiating with respect to x gives the slope of the tangent to the parabola at any point (x, y):
   dy/dx = d/dx (x² + 3x + 2)
dy/dx = 2x + 3

3. Find the point on the parabola where the tangent is parallel to the given line:
   For the shortest distance, the tangent to the parabola at the closest point must be parallel to the given line. This means their slopes must be equal.
   Set the derivative equal to the slope of the line:
   2x + 3 = 1
2x = 1 - 3
2x = -2
x = -1

4. Find the y-coordinate of this point on the parabola:
   Substitute x = -1 back into the parabola equation:
   y = (-1)² + 3(-1) + 2
y = 1 - 3 + 2
y = 0
   So, the point on the parabola closest to the line is P(-1, 0).

5. Calculate the perpendicular distance from the point P(-1, 0) to the line x - y - 3 = 0:
   The formula for the perpendicular distance d from a point (x₁, y₁) to a line Ax + By + C = 0 is:
   d = |Ax₁ + By₁ + C| / sqrt(A² + B²)
   Here, (x₁, y₁) = (-1, 0), A = 1, B = -1, and C = -3.
   d = |(1)(-1) + (-1)(0) - 3| / sqrt(1² + (-1)²)
d = |-1 + 0 - 3| / sqrt(1 + 1)
d = |-4| / sqrt(2)
d = 4 / sqrt(2)

6. Calculate d²:
   The question asks for d².
   d² = (4 / sqrt(2))²
d² = 16 / 2
d² = 8

The final answer is $\boxed{8}$.

Explanation of the solution:

The shortest distance between a line and a parabola (or any curve) occurs at a point on the curve where the tangent to the curve is parallel to the given line. We find this point by setting the derivative of the parabola's equation equal to the slope of the line. Once this point is found, the shortest distance is simply the perpendicular distance from this point to the given line.