Find the point on the line -2x + 4y + 3 =0 which is closest to the point (-2,1). Note: I have been struggling on this for hours!

Thank you very much Reiny for this wonderful explanation. I am not the brightest bulb and yes 2 hours to be precise.

To find the point on the line -2x + 4y + 3 = 0 that is closest to the point (-2, 1), we can use the concept of perpendicular distance.

Step 1: Rewriting the equation of the line
Let's rewrite the equation -2x + 4y + 3 = 0 in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

-2x + 4y + 3 = 0
4y = 2x - 3
y = (1/2)x - (3/4)

So, the equation of the line is y = (1/2)x - (3/4).

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

The given line has a slope of 1/2. Therefore, the slope of the line perpendicular to it is -2/1 or -2.

Step 3: Find the equation of the perpendicular line passing through the point (-2, 1)
We have the point (-2, 1) and the slope of the perpendicular line (-2). We can use the point-slope form of a line to find the equation of the perpendicular line.

The point-slope form of a line is y - y1 = m(x - x1), where (x1, y1) is a point on the line, and m is the slope.

Using the given point (-2, 1) and the slope -2, we substitute these values into the equation:

y - 1 = -2(x - (-2))
y - 1 = -2(x + 2)
y - 1 = -2x - 4
y = -2x - 3

So, the equation of the perpendicular line passing through (-2, 1) is y = -2x - 3.

Step 4: Find the point of intersection of the two lines
To find the point where the two lines intersect, we can solve the system of equations formed by the two line equations.

Setting the two equations equal to each other:

(1/2)x - (3/4) = -2x - 3

Simplifying:

5/2 x = -3 + 3/4
5/2 x = -12/4 + 3/4
5/2 x = -9/4

Multiplying both sides by 2/5 to solve for x:

x = (-9/4) * (2/5)
x = -18/20
x = -9/10

Substituting this value back into either equation to find y, we can use either the equation of the given line or the equation of the perpendicular line. Let's use the equation of the given line:

y = (1/2)x - (3/4)
y = (1/2)(-9/10) - (3/4)
y = -9/20 - 15/20
y = -24/20
y = -6/5

So, the point of intersection of the two lines is (-9/10, -6/5).

Step 5: Calculate the distance between the intersection point and (-2, 1)
To find the distance between two points, we can use the distance formula:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Using (-9/10, -6/5) as (x1, y1) and (-2, 1) as (x2, y2), we can calculate the distance:

d = sqrt((-2 - (-9/10))^2 + (1 - (-6/5))^2)
d = sqrt((-2 + 9/10)^2 + (1 + 6/5)^2)
d = sqrt((-20/10 + 9/10)^2 + (5/5 + 6/5)^2)
d = sqrt((-11/10)^2 + (11/5)^2)
d = sqrt((121/100) + (121/25))
d = sqrt(121/100 + 484/100)
d = sqrt(605/100)
d = sqrt(605)/10
d ≈ 7.78

Therefore, the point on the line -2x + 4y + 3 = 0 that is closest to the point (-2, 1) is approximately (-9/10, -6/5), with a distance of approximately 7.78 units.

To find the point on the line which is closest to the given point, we can use the concept of perpendicular distance.

The equation of the given line is: -2x + 4y + 3 = 0

We can rewrite this equation in the form of y = mx + c, where m is the slope of the line and c is the y-intercept. Rearranging the terms, we get:
4y = 2x - 3
y = (2/4)x - 3/4
y = (1/2)x - 3/4

Now, let's find the slope of the line perpendicular to this line. Perpendicular lines have negative reciprocal slopes. The slope of the given line is (1/2), so the slope of the perpendicular line will be -2.

Let's call the coordinates of the point we want to find on the line as (x1, y1). We need to find the x1 and y1 that satisfy both the equation of the given line and the equation of the perpendicular line.

Using the equation of the given line: y = (1/2)x - 3/4,
Substituting y by (1/2)x - 3/4 in the equation of the perpendicular line: y = -2x + c,
We get: (1/2)x - 3/4 = -2x + c

Now, we substitute the coordinates of the given point (-2,1) into this equation to find the value of c:
1 = -2(-2) + c
1 = 4 + c
c = -3

Now, we have the equation of the perpendicular line as y = -2x - 3.

To find the point (x1, y1) on the line -2x + 4y + 3 = 0, we substitute y1 by (-2x - 3) into this equation:
-2x + 4(-2x - 3) + 3 = 0

Solve this equation for x to find the x-coordinate of the point, and then substitute the x-coordinate back into the equation -2x + 4y + 3 = 0 to find the y-coordinate.

After finding the (x1, y1) coordinates, you can calculate the distance between this point and the given point (-2,1) using the distance formula:

Distance = sqrt((x1 - x2)^2 + (y1 - y2)^2)

where (x2, y2) represents the given point (-2,1).

I hope this explanation helps you understand the process of finding the point on the line closest to a given point and allows you to solve the problem more effectively.

For hours ???

Slope of the given line is 1/2
so the slope of the line through (-2,1) and perpendicular to it must be -2
equation of that perp.
y = -2x + b, with (-2,1) lying on it, thus ...
1 = -2(-2) + b
-3 = b

y = -2x - 3
solve the two equations to get the required point
-2x + 4y + 3 = 0
-2x + 4(-2x-3) + 3 = 0
-2x -8x - 12 + 3 = 0
-10x = 9
x = -9/10
sub that back into y = -2x-3
y = -2(-9/10) - 3 = -6/5

the point is (-9/10 , -6/5)