Coordinates of the Foot of the Perpendicular

To find the coordinates of the foot of the perpendicular from point (-1,3) to the line 3x-4y-16=0, we need to follow these steps:

1. Find the slope of the given line:
The equation of the line is given in the standard form Ax + By + C = 0. The slope-intercept form of a line is y = mx + b, where m is the slope. Rearranging the given equation, we get:
3x - 4y - 16 = 0
-4y = -3x + 16
y = (3/4)x - 4
Comparing this equation with the slope-intercept form, we can see that the slope of the line is 3/4.

2. Find the slope of the perpendicular line:
The slope of a line perpendicular to another line is the negative reciprocal of its slope. Therefore, the slope of the perpendicular line is -4/3.

3. Use the point-slope form 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. We can use the given point (-1,3) and the slope -4/3 to find the equation of the perpendicular line:
y - 3 = (-4/3)(x - (-1))
y - 3 = (-4/3)(x + 1)
y - 3 = (-4/3)x - 4/3
y = (-4/3)x - 4/3 + 3
y = (-4/3)x - 4/3 + 9/3
y = (-4/3)x + 5/3
Simplifying the equation, we get:
4x + 3y - 5 = 0

4. Find the intersection point of the two lines:
To find the foot of the perpendicular, we need to find the intersection point of the given line and the perpendicular line. We can do this by solving the system of equations formed by the two lines:
3x - 4y - 16 = 0
4x + 3y - 5 = 0
Solving these equations simultaneously, we find that the intersection point is (1, 2).

Distance of Point (1,3) from the Line

To find the distance of point (1,3) from the line 3x-4y-16=0, we can use the formula for the distance between a point and a line.

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

In this case, the point (x1, y1) is (1, 3) and the equation of the line is 3x - 4y - 16 = 0.

Substituting the values into the formula, we get:
Distance = |3