Find the point on the line 4x+3y-4=0 which is closest to the point (5,2)

Thank you for your help!

To find the point on the line 4x + 3y - 4 = 0, which is closest to the point (5,2), we need to use the formula for the distance between a point and a line.

Step 1: Convert the line equation into slope-intercept form (y = mx + b).
4x + 3y - 4 = 0
3y = -4x + 4
y = (-4/3)x + 4/3

Step 2: Use the formula for the distance between a point (x1, y1) and a line (Ax + By + C = 0):
Distance = |Ax1 + By1 + C| / sqrt(A^2 + B^2)

Let's plug in the values:
A = 4, B = 3, C = -4, x1 = 5, y1 = 2

Distance = |4(5) + 3(2) - 4| / sqrt(4^2 + 3^2)
Distance = |20 + 6 - 4| / sqrt(16 + 9)
Distance = |22| / sqrt(25)
Distance = 22 / 5

Step 3: To find the point on the line that is closest to (5,2), we need to find the foot of the perpendicular from (5,2) to the line.

The slope of the perpendicular line will be the negative reciprocal of the slope of the given line.

The slope of the given line is -4/3, so the slope of the perpendicular line will be 3/4.

Using the point-slope form of a line (y - y1) = m(x - x1), we can find the equation of the perpendicular line passing through (5,2):
(y - 2) = (3/4)(x - 5)

Let's simplify the equation:
4(y - 2) = 3(x - 5)
4y - 8 = 3x - 15
4y = 3x - 7
y = (3/4)x - 7/4

The point of intersection of the two lines (the given line and the perpendicular line) will be the point on the given line that is closest to (5,2).

Now, we can solve the system of equations:
y = (-4/3)x + 4/3
y = (3/4)x - 7/4

Substituting the first equation into the second equation:
(-4/3)x + 4/3 = (3/4)x - 7/4

Now, solve for x:
Multiply through by 12 to eliminate the fractions:
-16x + 16 = 9x - 21

Collect the x terms:
-16x - 9x = -21 - 16
-25x = -37
x = -37 / -25
x = 37/25

Now, substitute x = 37/25 back into y = (-4/3)x + 4/3 to find y:
y = (-4/3)(37/25) + 4/3
y = -148/75 + 100/75
y = -48/75
y = -16/25

Therefore, the point on the line 4x + 3y - 4 = 0, which is closest to the point (5,2), is (37/25, -16/25).

To find the point on the line 4x + 3y - 4 = 0 that is closest to the point (5,2), we can use the concept of distance between two points.

Step 1: Identify the equation of the given line.
The equation of the given line is 4x + 3y - 4 = 0.

Step 2: Find the perpendicular line passing through the point (5,2).
Since we want to find the point on the line that is closest to the given point, we need to find the perpendicular line passing through (5,2). The slope of the given line is -4/3, so the slope of the perpendicular line will be the negative reciprocal, which is 3/4. Using the point-slope form equation, we can find the equation of the perpendicular line:
y - 2 = (3/4)(x - 5)
4y - 8 = 3x - 15
3x - 4y + 7 = 0

Step 3: Solve the system of equations formed by the given line and the perpendicular line.
Now we need to solve the system of equations formed by the given line 4x + 3y - 4 = 0 and the perpendicular line 3x - 4y + 7 = 0. We can solve this system using any method, such as substitution or elimination.

Let's use the elimination method to solve it:
Multiply the first equation by 3 and the second equation by 4 to eliminate the y term:
12x + 9y - 12 = 0
12x - 16y + 28 = 0

Now subtract the second equation from the first:
0 - 25y - 40 = 0
-25y = 40
y = -40/25
y = -8/5

Substitute the value of y into either one of the original equations and solve for x:
4x + 3(-8/5) - 4 = 0
4x - 24/5 - 20/5 = 0
4x - 44/5 = 0
4x = 44/5
x = (44/5) * (1/4)
x = 11/5

Step 4: Find the point on the line.
The point on the line 4x + 3y - 4 = 0 that is closest to the point (5,2) is (11/5, -8/5).

A line from (5,2) perpendicular to the curve will have the shortest distance to the curve.

The slope of the graph is -4/3, so the normal has slope 3/4

So, now you have a point and a slope, so the perpendicular line through (5,2) is

y-2 = 3/4 (x-5)
y = 3/4 x - 7/4

These two lines intersect at (37/25,-16/25)