Final answer:
To find the first quadrant point on the curve y^2 = 18x that is closest to the point (2, 0), we can use the distance formula and calculus. Rearrange the equation to get x in terms of y, calculate the distance between the two points, and take the derivative to find the y-coordinate of the closest point. Substitute the y-coordinate into the equation to find the x-coordinate.
Step-by-step explanation:
We can rearrange the equation y^2 = 18x to get x = (y^2)/18. To find the point on the curve that is closest to (2, 0), we need to minimize the distance between the point (2, 0) and any general point (x, y) on the curve. We can use the distance formula to calculate the distance between the two points:
d = sqrt((x - 2)^2 + (y - 0)^2)
We substitute x = (y^2)/18 into the distance formula and take the derivative of d with respect to y. Setting the derivative equal to zero will give us the y-coordinate of the closest point:
d/dy(sqrt(((y^2)/18 - 2)^2 + (y - 0)^2)) = 0
Solving this equation will give us the y-coordinate of the closest point, and then we can substitute it back into x = (y^2)/18 to find the x-coordinate.