Find the point on the parabola y^2=2x that is closest to the point (2,4)
To find the point on the parabola y^2=2x that is closest to the point (2,4), we can use a method called the distance formula.
1. Start by finding the general equation of the parabola. Rewrite the equation y^2 = 2x as x = (y^2)/2.
2. Set up the distance formula, which is the square root of the sum of the squares of the differences in the x-coordinates and y-coordinates of the two points. Let's call the point on the parabola (x, y).
The distance formula is: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, (x1, y1) is (2, 4) and (x2, y2) is (x, y).
So, the distance formula becomes: d = sqrt((x - 2)^2 + (y - 4)^2).
3. Notice that we can rewrite x = (y^2)/2 in terms of y as: x = y^2/2.
4. Substitute x = y^2/2 into the distance formula, so we get: d = sqrt((y^2/2 - 2)^2 + (y - 4)^2).
5. Simplify the equation by expanding and combining like terms: d = sqrt((y^4/4 - 2y^2 + 4) + (y^2 - 8y + 16)).
6. Further simplify the equation: d = sqrt(y^4/4 - 2y^2 + 4 + y^2 - 8y + 16).
7. Combine like terms: d = sqrt(y^4/4 - y^2 - 8y + 20).
8. Take the derivative of the equation with respect to y to find the minimum distance. Set d' = 0 and solve for y.
d' = (1/4) * (4y^3 - 2y - 8) = 0.
Simplify the equation: 4y^3 - 2y - 8 = 0.
Use a numerical method like the Newton-Raphson method or graphing calculator to find the approximate solution(s) for y.
9. After finding the value(s) of y, substitute them back into the equation x = y^2/2 to obtain the corresponding x-values.
This will give you the coordinate(s) of the point(s) on the parabola closest to the point (2, 4).