Final answer:
The closest point on the line 4x - 2y - 5 = 0 to the point (5,0) is found by solving the equations of the original line and a perpendicular line that passes through (5,0). The closest point is (2, 3/2).
Step-by-step explanation:
To find the point on the line 4x - 2y - 5 = 0 that is closest to the point (5,0), we need to minimize the distance between the point and the line. The standard way to do this is to use the formula for the distance from a point to a line, but herein we will utilize the fact that the perpendicular distance from a point to a line is the shortest distance.
The slope of the line 4x - 2y - 5 = 0 is 2, so the slope of the perpendicular line will be the negative reciprocal, which is -1/2. The equation of the line that passes through the point (5,0) with this slope can be expressed as y + 1/2x = 5/2.
We find the intersection point of these two lines by simultaneously solving their equations:
- Rewrite the line 4x - 2y - 5 = 0 in slope-intercept form: y = 2x - 5/2.
- Set the two equations equal to each other: 2x - 5/2 = -1/2x + 5/2.
- Solve for x: 2x + 1/2x = 5, x = 2.
- Substitute x into one of the original equations to find y: y = 2x - 5/2, y = 4 - 5/2, y = 3/2.
Therefore, the point on the line that is closest to (5,0) is (2, 3/2).