A line with negative slope passes through the point P(5,2) and intersects the x-axis at the point A and the y-axis at the point B.
a. Find the equation of the line for which the length of the segment AB
is the smallest.
b. Find the equation of the line for which sum of the lengths OA +OB
the smallest.
c. Find the equation of the line for which the area of the triangle OAB
is the smallest.
To find the equation of the line for each scenario, we will utilize the point-slope form of a linear equation, which is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line, and m is the slope of the line.
a. To find the line that minimizes the length of segment AB, we need to find the line that passes through point P(5,2), intersects the x-axis at point A, and intersects the y-axis at point B.
The slope of a line with negative slope is negative. Since the line intersects the x-axis at point A, the y-coordinate of A will be 0, and the x-coordinate will be some value x.
We can use the point-slope form to find the equation of the line:
y - 2 = -m(x - 5)
To find the equation, we need to find the slope m.
Since the line passes through points P(5,2) and A(x,0), we can use the slope formula:
m = (y2 - y1) / (x2 - x1)
m = (0 - 2) / (x - 5)
Given that the slope is negative, we can substitute the value of m into the equation:
y - 2 = -[(0 - 2) / (x - 5)](x - 5)
Simplifying this equation will give us the equation of the line that minimizes the length of segment AB.
b. To find the line that minimizes the sum of the lengths OA + OB, we need to find the line that passes through point P(5,2) and intersects both the x-axis and the y-axis.
Since the line passes through point P(5,2), the equation can be written as:
y - 2 = mx - 5m
To find the equations of the x and y-intercepts, we need to substitute the appropriate values into the equation.
When the line intersects the x-axis, the y-coordinate is 0, so we have:
0 - 2 = mx - 5m
-2 = x(m - 5)
When the line intersects the y-axis, the x-coordinate is 0, so we have:
y - 2 = 0 - 5m
y = -5m + 2
By determining the values of x and y, we can obtain the equation of the line that will minimize the sum of the lengths OA + OB.
c. To find the line that minimizes the area of triangle OAB, we need to find the line that passes through points P(5,2) and intersects both the x-axis and the y-axis.
Since the triangle OAB is formed from the x and y-intercepts, the equation of the line can be written as:
y - 2 = mx - 5m
To find the equations of the x and y-intercepts, we can substitute appropriate values into the equation.
When the line intersects the x-axis, the y-coordinate is 0, so we have:
0 - 2 = mx - 5m
-2 = x(m - 5)
When the line intersects the y-axis, the x-coordinate is 0, so we have:
y - 2 = 0 - 5m
y = -5m + 2
By determining the values of x and y, we can obtain the equation of the line that will minimize the area of triangle OAB.