Problem Statement:
A straight line passing through point P(3,1) meets the coordinate axes at points A and B. The objective is to determine the maximum distance of the straight line from the origin O and calculate the area of triangle OAB.

Solution:

1. Finding the equation of the straight line:
Let the equation of the straight line passing through point P(3,1) be y = mx + c, where m is the slope and c is the y-intercept.

Since the line passes through point P(3,1), we can substitute these coordinates into the equation to get:
1 = 3m + c ...(1)

2. Finding the coordinates of points A and B:
To find the coordinates of points A and B, we substitute the x and y values of each point into the equation of the line.

For point A, we have:
x = 0 (since it lies on the y-axis)
0 = 3m + c

For point B, we have:
y = 0 (since it lies on the x-axis)
0 = mx + c

Solving these two equations simultaneously, we get the values of m and c.

3. Calculating the distance from origin O:
The distance between a point (x1, y1) and the origin O(0,0) is given by the formula:
distance = √[(x1 - 0)^2 + (y1 - 0)^2]
= √[x1^2 + y1^2]

We can calculate the distance from origin O for any point on the line using this formula.

4. Finding the maximum distance:
To find the maximum distance of the line from the origin O, we need to maximize the distance function. Since the square root function is strictly increasing, we can maximize the square of the distance instead.

Let D be the square of the distance from the origin O. We have:
D = x^2 + y^2

Substituting the values of x and y from the equation of the line, we get:
D = (3m + c)^2 + 1^2
= 9m^2 + 6mc + c^2 + 1

5. Maximizing the distance function:
To find the maximum value of D, we can differentiate the function with respect to m and c, and equate the derivatives to zero.

dD/dm = 18m + 6c = 0 ...(2)
dD/dc = 6m + 2c = 0 ...(3)

Solving equations (2) and (3) simultaneously, we can find the values of m and c that maximize the distance function.

6. Calculating the coordinates of points A and B:
Once we have the values of m and c, we can substitute them back into the equation of the line to find the coordinates of points A and B.

7. Calculating the area of triangle OAB:
The area of a triangle can be calculated using the formula:
Area = 0.5 * base * height

In this case, the length of the base is the distance between points A and B, and the height is the distance from point O