Final answer:
To find the closest point on the line 2x + 4y + 4 = 0 to the point (2, 4), calculate the perpendicular distance from the point to the line, and determine where this perpendicular line intersects the original line.
Step-by-step explanation:
The student's question seeks to find the point on the line 2x + 4y + 4 = 0 that is closest to the point (2, 4). To answer this, we first need to calculate the perpendicular distance from the given point to the line, as the closest point on the line to the point (2, 4) will be where this perpendicular intersects the line.
First, we'll express the equation of the line in slope-intercept form to identify its slope. We'll then use this slope to determine the slope of the perpendicular line, which is the negative reciprocal of the original line's slope. The equation of our line in slope-intercept form is y = -x/2 - 1. Therefore, the slope of the perpendicular line is 2 since the slope of the original line is -1/2. Using point-slope form, we can form an equation for the line perpendicular to the original one that passes through the point (2, 4).
The final step is just algebra. We need to find the intersection point between the original line and the perpendicular line we formed. Calculating this intersection will give us the closest point to (2, 4) on the line 2x + 4y + 4 = 0.