The point on the parabola y=x^2 that is closest to the point (2,1/2) is (___,___). The distance between the two points is _______.
To find the point on the parabola y = x^2 that is closest to the point (2, 1/2), we need to find the point on the parabola where the distance between the two points is minimized.
Step 1: Find the distance between the two points
The distance between two points (x1, y1) and (x2, y2) is given by the formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, the points are (x, x^2) and (2, 1/2). Let's plug these values into the formula:
Distance = sqrt((2 - x)^2 + (1/2 - x^2)^2)
Step 2: Minimize the distance function
To minimize the distance function, we can take the derivative of the distance function with respect to x and set it equal to 0. Then, solve for x.
dy/dx = 0
Step 3: Calculate the coordinates of the closest point
Once we find the value of x, we can substitute it back into the equation y = x^2 to find the y-coordinate of the closest point on the parabola.
The coordinates of the closest point on the parabola y = x^2 are (x, x^2). The distance between the two points can be calculated using the distance formula.
The distance z from (2,1/2) to (x,x^2) is
z = √[(x-2)^2 + (x^2 - 1/2)^2]
= 1/2 √(4x^4-16x+17)
dz/dx = 4(x^3-1)/√(4x^4-16x+17)
dz/dx=0 at x=1
z(1) = √5/2
Or, consider the line through (x,x^2) and (2,1/2). It has slope
(1/2 - x^2)/(2-x)
The tangent line to y=x^2 has slope 2x. So, we want the two lines to be perpendicular. That means
(1/2 - x^2)/(2-x) * 2x = -1
x=1 as above