Find the point on the line 2x+4y+7=0 which is closest to the point (4,−3).

the distance from (x,y) to (4,-3) is

√((x-4)^2 + (y+3)^2)
= √((x-4)^2 + ((2x+7)/4+3)^2)
= √5/4 (4x^2-20x+61)
This has a minimum at x = 37/10

So, the point is (37/10,-18/5)

Or, you can consider the perpendicular line through (4,-3). It is
y+3 = 2 (x-4)
This line intersects the other where

2(x-4)-3 = -(2x+7)/4
x = 37/10
. . .

To find the point on the line 2x + 4y + 7 = 0 that is closest to the point (4, -3), we can use the concept of perpendicular distance.

Step 1: Write the given equation in slope-intercept form (y = mx + b):
2x + 4y + 7 = 0
4y = -2x - 7
y = (-1/2)x - 7/4

Step 2: Determine the slope (m) of the given line:
The slope of the given line is -1/2.

Step 3: Find the negative reciprocal of the slope to get the slope of the line perpendicular to the given line:
The slope of the line perpendicular to the given line is 2.

Step 4: Use the formula for the distance between a point (x₁, y₁) and a line with equation Ax + By + C = 0:
The distance, d, between a point (x₁, y₁) and a line with equation Ax + By + C = 0 is given by:
d = |Ax₁ + By₁ + C| / (√(A² + B²))

Using point (4, -3) and the equation 2x + 4y + 7 = 0, we can find the perpendicular distance.

Step 5: Substitute the coordinates (x₁, y₁) = (4, -3) into the equation and calculate:
d = |2(4) + 4(-3) + 7| / (√(2² + 4²))
= |8 - 12 + 7| / (√(4 + 16))
= |-4 + 7| / (√20)
= 3 / (√20)

Step 6: Simplify the expression (√20):
(√20) = (√(4 × 5)) = (√4 × √5) = 2√5

Step 7: Substitute the simplified expression back into the distance formula:
d = 3 / (2√5)

So, the closest point on the line 2x + 4y + 7 = 0 to the point (4, -3) has a perpendicular distance of 3 / (2√5) from the given point.