Question

Prove that the distance between (at^2,2at) and (at^2,2at) is zero.

a. The points are at the same location.
b. The distance cannot be determined.
c. The distance is always positive.
d. The points are not on the same vertical line.

Answer 1

The distance between the points (at^2, 2at) and (at^2, 2at) is zero because they have identical coordinates, which means they are the same point.

Step-by-step explanation:

The distance between two points (at^2, 2at) and (at^2, 2at) is given by the formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\), where (x_1, y_1) and (x_2, y_2) are the coordinates of the two points. Since both points have identical coordinates, the difference between their x-coordinates and y-coordinates will both be zero. Therefore, the distance is zero as the formula simplifies to \(d = \sqrt{0^2 + 0^2} = 0\).

In this case, the correct answer is a. The points are at the same location. There is no need to measure or calculate the distance further as both points share the exact same coordinates, indicating that they are one and the same point. Hence, the distance cannot be anything other than zero.