Answer:
(2, 2)
Step-by-step explanation:
According to the distance formula, the distance between two points is:
d² = (x₂ − x₁)² + (y₂ − y₁)²
If one point is (x, y) and the other point is (1, 4), then:
d² = (x − 1)² + (y − 4)²
We know y² = 2x, so x = ½ y². Substituting:
d² = (½ y² − 1)² + (y − 4)²
The minimum distance is when dd/dy equals 0. We can either simplify first by distributing, or we can immediately take the derivative using chain rule.
If we distribute and then take the derivative:
d² = ¼ y⁴ − y² + 1 + y² − 8y + 16
d² = ¼ y⁴ − 8y + 17
2d dd/dy = y³ − 8
If we use chain rule instead without distributing:
2d dd/dy = 2(½ y² − 1) (y) + 2(y − 4)
2d dd/dy = y³ − 2y + 2y − 8
2d dd/dy = y³ − 8
Setting dd/dy equal to 0:
0 = y³ − 8
y = 2
x = ½ y²
x = 2
(2, 2) is the point on the parabola closest to (1, 4).
Graph: desmos.com/calculator/m4apqwsduk
