To find the point on the parabola y² = 2x that is closest to the point (1, 4), we can use the method of minimizing the distance between a point on the parabola and the given point.
First, we express the distance D between a point (x, y) on the parabola and the point (1, 4) using the distance formula:
D = √((x - 1)² + (y - 4)²)Since we want to minimize the distance, it’s easier to minimize the square of the distance (D²):
D² = (x - 1)² + (y - 4)²Now, substitute y in terms of x from the parabola’s equation. From y² = 2x, we have:
y = ±√(2x)For simplification, we can work with the positive branch of the parabola first. Thus, substituting y gives us:
D² = (x - 1)² + (√(2x) - 4)²We need to minimize this function with respect to x. Expanding the expression results in:
D² = (x - 1)² + (2x - 8√(2x) + 16)Next, we can combine the terms:
D² = x² - 2x + 1 + 2x - 8√(2x) + 16Thus, we have:
D² = x² - 8√(2x) + 17To find the minimum, we will take the derivative of D² with respect to x and set it to zero:
d(D²)/dx = 2x - (8 * (1/√(2x)) * (1/2√(2))) = 2x - 4√(2/x)Setting the derivative equal to zero:
2x - 4√(2/x) = 0Solving for x, we rearrange:
2x = 4√(2/x)
√(2/x) = x/2Square both sides to eliminate the square root:
2/x = x²/4
Multiplying through by 4x gives:
8 = x³This leads to:
x = 2Now, substitute this value back into y² = 2x to find y:
y² = 2(2) = 4Thus, y = ±2. We have two possible points on the parabola: (2, 2) and (2, -2).
Next, we compare the distances of these two points to (1, 4):
For point (2, 2):
D = √((2 - 1)² + (2 - 4)²) = √(1 + 4) = √5For point (2, -2):
D = √((2 - 1)² + (-2 - 4)²) = √(1 + 36) = √37Since √5 (approximately 2.24) is less than √37 (approximately 6.08), the point on the parabola y² = 2x that is closest to the point (1, 4) is:
(2, 2)