# Find the slope and y-intercept of the line through the point (6,2) that cuts off the least area from the first quadrant.

## y = m x + b

2 = 6 m + b
m=(2-b)/6

when y = 0
x = -b/m
so
Area = (1/2)b(-b/m)
A = -b^2/2m
A = -3b^2/(2-b)
dA/db = [(2-b)(-6b)+3b^2(-1)]/bot^2
=0 for min or max
3 b^2 = -12b + 6 b^2
3 b^2 = 12 b
b = 4 (or zero of course)
so y intercept is 4
m = (2-b)/6
= -2/6 = -1/3

## To find the slope and y-intercept of the line through the point (6,2) that cuts off the least area from the first quadrant, we first need to understand the concept of "cutting off the least area."

When a line passes through a point in the first quadrant, it divides the first quadrant into two regions. These regions are triangular in shape, with the line forming one of the sides. The area of the triangular region is given by the formula A = (1/2)bh, where b is the length of the line segment that is parallel to the x-axis, and h is the height of the triangle.

Since we want to find the line that cuts off the least area, we need to minimize the product bh, or equivalently, minimize both b and h.

The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. We can plug in the coordinates (6,2) into this equation to find the slope and y-intercept.

Using the point (6,2), we have:
2 = m * 6 + b

To find the slope, we need another point on the line. Since we know that the line passes through the point (6,2), we can choose any other point on the line. Let's choose (0, b) as another point.

Using the points (6,2) and (0, b), we can calculate the slope by using the formula: m = (y2 - y1) / (x2 - x1)

m = (b - 2) / (0 - 6)
m = (b - 2) / (-6)

Now we have two equations:
2 = 6m + b
m = (b - 2) / (-6)

We can solve these two equations simultaneously to find the slope and y-intercept.

Rearranging the first equation, we have:
b = 2 - 6m

Substituting this value of b into the second equation:
m = ((2 - 6m) - 2) / (-6)
m = (2 - 6m - 2) / (-6)
m = -4m / -6

Simplifying, we get:
m = 2/3

Substituting this value of m into the first equation:
2 = 6 * (2/3) + b
2 = 4 + b
b = -2

Therefore, the slope of the line is 2/3 and the y-intercept is -2.

## To find the slope and y-intercept of the line that cuts off the least area from the first quadrant, we need to consider the line that is perpendicular to the line passing through the point (6,2) and intersects the x-axis at the point (a, 0). The reason for considering this perpendicular line is that it will cut off the least area from the first quadrant.

Step 1: Find the slope of the line passing through (6,2)
The slope of a line passing through two points (x_1, y_1) and (x_2, y_2) is given by the formula:
slope = (y_2 - y_1) / (x_2 - x_1)

In this case, we have (x_1, y_1) = (6, 2) and (x_2, y_2) = (a, 0). Substituting the values into the formula, we get:
slope = (0 - 2) / (a - 6) = -2 / (a - 6)

Step 2: Find the slope of the perpendicular line
The slope of a perpendicular line to a given line is the negative reciprocal of the slope of the given line. In other words, if the slope of the given line is m, then the slope of the perpendicular line is -1/m.

In this case, the slope of the perpendicular line is -1 / (-2 / (a - 6)) = (a - 6) / 2.

Step 3: Find the y-intercept of the perpendicular line
The y-intercept is the point where the line intersects the y-axis. Since the line passes through the point (a, 0), the y-intercept is 0.

Therefore, the equation of the perpendicular line is y = (a - 6) / 2 * x.

Step 4: Finding the line that cuts off the least area from the first quadrant
In order to find the line that cuts off the least area from the first quadrant, we want to minimize the area of the triangle formed by the line passing through (6, 2) and the x and y axes.

The area of the triangle formed by the line passing through (6, 2) and the x and y axes is given by the formula:
area = (1/2) * (base) * (height)

In this case, the base of the triangle is the x-intercept, which is the point where the line intersects the x-axis. To find the x-intercept, we set y to 0 and solve for x:

0 = (a - 6) / 2 * x

Simplifying the equation, we get:
0 = (a - 6) * x

Since we want to find the line that cuts off the least area, we want to minimize the base of the triangle. The smallest possible value for the base occurs when x = 0.

Therefore, the x-intercept of the line is (0, 0).

The slope of the line passing through (6, 2) that cuts off the least area from the first quadrant is (a - 6) / 2, and the y-intercept is 0.